💡First, calculate the original total sum of the 50 observations using the incorrect mean:[text{Original Sum} = 30 times 50 = 1500]Next, determine the difference caused by the incorrect observation:[text{Difference} = 48 - 23 = 25]Add this difference to the original sum to find the corrected total sum:[text{Corrected Sum} = 1500 + 25 = 1525]Finally, calculate the corrected mean by dividing the corrected sum by the number of observations:[text{Corrected Mean} = frac{1525}{50} = 30.5]

💡Alright, so I have this problem here: Given five numbers, the sum of any three of them is even. I need to prove that all the numbers are even. Hmm, okay, let me think about this step by step.First, let's recall that when we talk about the sum of numbers being even, it relates to the parity of the numbers. Parity refers to whether a number is even or odd. So, if I have three numbers, their sum is even. I need to figure out what that implies about each individual number.Let me denote the five numbers as A, B, C, D, and E. The problem states that the sum of any three of these is even. That means:- A + B + C is even- A + B + D is even- A + B + E is even- A + C + D is even- A + C + E is even- A + D + E is even- B + C + D is even- B + C + E is even- B + D + E is even- C + D + E is evenThat's a lot of sums! Maybe I can find a pattern or relationship between these sums to figure out the parity of each number.Let me start by looking at the first two sums: A + B + C and A + B + D. Both are even. If I subtract these two sums, I get (A + B + C) - (A + B + D) = C - D. Since both sums are even, their difference should also be even. So, C - D is even, which means C and D have the same parity. In other words, both C and D are either even or both are odd.Similarly, if I take another pair of sums, say A + B + C and A + C + D, both are even. Subtracting them gives (A + B + C) - (A + C + D) = B - D. Again, since both sums are even, their difference is even, so B - D is even. This means B and D have the same parity.Wait a minute, so B and D have the same parity, and C and D have the same parity. That means B, C, and D all have the same parity. So, they are either all even or all odd.Let me check another pair. How about A + B + C and A + C + E? Both are even. Subtracting them gives (A + B + C) - (A + C + E) = B - E. So, B - E is even, meaning B and E have the same parity. But we already know B has the same parity as D and C, so E must also have the same parity as B, C, and D.So far, I have that B, C, D, and E all have the same parity. Let's see about A. Take the sums A + B + C and A + B + D. Both are even, and we know B, C, and D have the same parity. Let's assume, for a moment, that B, C, D, and E are all even. Then, what does that say about A?If B, C, and D are even, then A + B + C is even. Since B and C are even, their sum is even, so A must also be even to make the total sum even. Similarly, A + B + D is even, and since B and D are even, A must be even as well.Alternatively, if B, C, D, and E are all odd, then let's see what happens. If B, C, and D are odd, then A + B + C would be A + (odd + odd) = A + even. For this sum to be even, A must be even. Similarly, A + B + D would be A + (odd + odd) = A + even, so A must be even. But wait, if A is even and B, C, D, E are odd, let's check another sum. How about A + C + E? That would be even + odd + odd = even + even = even, which is fine. Similarly, A + D + E would be even + odd + odd = even + even = even. Okay, that still works.But hold on, let's look at the sum B + C + D. If B, C, and D are all odd, then their sum is odd + odd + odd = odd + even = odd. But the problem states that the sum of any three numbers is even. So, B + C + D must be even. But if B, C, and D are all odd, their sum is odd, which contradicts the given condition.Therefore, our assumption that B, C, D, and E are all odd must be wrong. So, they must all be even. Therefore, A must also be even, as we saw earlier.So, putting it all together, since B, C, D, and E are all even, and A is even as well, all five numbers must be even.Wait, let me double-check to make sure I didn't miss anything. I considered the parity of the sums and used the fact that the difference of two even numbers is even. I concluded that B, C, D, and E must have the same parity, and then checked the implications for A. Then, I considered both possibilities for the parity of B, C, D, and E and found that if they were odd, it led to a contradiction. Therefore, they must all be even, and so must A.I think that covers all the cases. I don't see any gaps in the logic. So, yeah, all five numbers must be even.**Final Answer**boxed{text{All five numbers are even.}}

💡Okay, so I have this geometry problem about an ellipse. Let me try to visualize it first. There's an ellipse with major axis AB, center O, and a focus F. Point P is somewhere on the ellipse. Then there's a chord CD passing through O and parallel to the tangent at P. The line PF intersects CD at point Q. I need to prove or disprove that PQ equals OA, which is the semi-major axis length.Alright, let's break this down. First, I should recall some properties of ellipses. An ellipse has two foci, and the sum of the distances from any point on the ellipse to the two foci is constant and equal to the major axis length, which is 2a. So, OA and OB are both equal to a, since O is the center.Now, CD is a chord through O, so it's a diameter of the ellipse. Also, CD is parallel to the tangent at P. That seems important. I remember that the tangent at a point on an ellipse has a specific slope related to the point's coordinates. Maybe I can use that.Let me try to sketch this in my mind. The ellipse is symmetric about both axes. The major axis is AB, so O is the midpoint. F is one focus, and the other focus, let's call it F', is on the opposite side of O. Point P is somewhere on the ellipse, not necessarily on the major axis. The tangent at P is a line that just touches the ellipse at P. CD is a chord through O and parallel to this tangent.So, if CD is parallel to the tangent at P, then CD must have the same slope as the tangent at P. That might help me relate the positions of P and CD.Next, PF is a line from P to focus F, and it intersects CD at Q. I need to find the length PQ and see if it's equal to OA, which is a.Hmm, maybe I can use coordinate geometry here. Let's set up a coordinate system with O at the origin. Let me denote the major axis AB along the x-axis. So, O is (0,0), A is (-a,0), B is (a,0), and F is at (c,0), where c is the distance from the center to the focus. For an ellipse, c^2 = a^2 - b^2, where b is the semi-minor axis.Point P is on the ellipse, so its coordinates can be represented parametrically. Let me use the standard parametric equations for an ellipse: P = (a cos θ, b sin θ), where θ is the parameter.The tangent at P has a specific equation. For an ellipse, the equation of the tangent at point (a cos θ, b sin θ) is (x cos θ)/a + (y sin θ)/b = 1. So, the slope of this tangent is - (b cos θ)/(a sin θ).Since CD is parallel to this tangent, CD must have the same slope. CD is a chord passing through O, so its equation is y = m x, where m is the slope, which is - (b cos θ)/(a sin θ).So, the equation of CD is y = [ - (b cos θ)/(a sin θ) ] x.Now, I need to find the equation of line PF. Point P is (a cos θ, b sin θ), and F is (c,0). So, the slope of PF is (b sin θ - 0)/(a cos θ - c) = (b sin θ)/(a cos θ - c).Thus, the equation of PF is y = [ (b sin θ)/(a cos θ - c) ] (x - c).Now, point Q is the intersection of PF and CD. So, let's set the two equations equal:[ (b sin θ)/(a cos θ - c) ] (x - c) = [ - (b cos θ)/(a sin θ) ] x.Let me solve for x.Multiply both sides by (a cos θ - c) * (a sin θ) to eliminate denominators:b sin θ * a sin θ * (x - c) = -b cos θ * (a cos θ - c) * x.Simplify:a b sin² θ (x - c) = -b cos θ (a cos θ - c) x.Divide both sides by b:a sin² θ (x - c) = -cos θ (a cos θ - c) x.Expand the left side:a sin² θ x - a sin² θ c = -a cos² θ x + c cos θ x.Bring all terms to one side:a sin² θ x - a sin² θ c + a cos² θ x - c cos θ x = 0.Factor x:x (a sin² θ + a cos² θ - c cos θ) - a sin² θ c = 0.Note that sin² θ + cos² θ = 1, so:x (a - c cos θ) - a sin² θ c = 0.Solve for x:x = [ a sin² θ c ] / (a - c cos θ).So, x-coordinate of Q is [ a c sin² θ ] / (a - c cos θ).Now, let's find the y-coordinate using the equation of CD: y = [ - (b cos θ)/(a sin θ) ] x.Substitute x:y = [ - (b cos θ)/(a sin θ) ] * [ a c sin² θ / (a - c cos θ) ].Simplify:y = - (b cos θ / a sin θ) * (a c sin² θ / (a - c cos θ)).The a cancels, sin θ cancels with sin² θ leaving sin θ:y = - (b cos θ) * (c sin θ / (a - c cos θ)).So, y = - (b c cos θ sin θ) / (a - c cos θ).Therefore, coordinates of Q are:( [ a c sin² θ ] / (a - c cos θ), - (b c cos θ sin θ) / (a - c cos θ) ).Now, I need to find the distance PQ.Point P is (a cos θ, b sin θ).Point Q is ( [ a c sin² θ ] / (a - c cos θ), - (b c cos θ sin θ) / (a - c cos θ) ).So, let's compute the differences in coordinates.Δx = [ a c sin² θ / (a - c cos θ) ] - a cos θ.Δy = [ - (b c cos θ sin θ ) / (a - c cos θ) ] - b sin θ.Let me compute Δx first:Δx = [ a c sin² θ - a cos θ (a - c cos θ) ] / (a - c cos θ).Expand the numerator:a c sin² θ - a² cos θ + a c cos² θ.Factor a:a [ c sin² θ - a cos θ + c cos² θ ].Note that sin² θ + cos² θ = 1, so:a [ c (sin² θ + cos² θ) - a cos θ ] = a [ c - a cos θ ].Thus, Δx = [ a (c - a cos θ) ] / (a - c cos θ).Similarly, compute Δy:Δy = [ - b c cos θ sin θ - b sin θ (a - c cos θ) ] / (a - c cos θ).Factor out -b sin θ:Δy = [ -b sin θ (c cos θ + a - c cos θ) ] / (a - c cos θ).Simplify inside the parentheses:c cos θ + a - c cos θ = a.Thus, Δy = [ -b sin θ * a ] / (a - c cos θ).So, Δy = - a b sin θ / (a - c cos θ).Now, let's compute PQ distance:PQ = sqrt( (Δx)^2 + (Δy)^2 ).Compute (Δx)^2:[ a (c - a cos θ) / (a - c cos θ) ]^2 = [ a ( - (a cos θ - c) ) / (a - c cos θ) ]^2 = [ -a (a cos θ - c) / (a - c cos θ) ]^2.But since it's squared, the negative goes away:= [ a (a cos θ - c) / (a - c cos θ) ]^2.Similarly, (Δy)^2 = [ - a b sin θ / (a - c cos θ) ]^2 = [ a b sin θ / (a - c cos θ) ]^2.So, PQ^2 = [ a^2 (a cos θ - c)^2 + a^2 b^2 sin^2 θ ] / (a - c cos θ)^2.Factor out a^2 in the numerator:= a^2 [ (a cos θ - c)^2 + b^2 sin^2 θ ] / (a - c cos θ)^2.Let me compute the numerator inside the brackets:(a cos θ - c)^2 + b^2 sin^2 θ.Expand (a cos θ - c)^2:= a^2 cos² θ - 2 a c cos θ + c^2 + b^2 sin² θ.Now, recall that for an ellipse, c^2 = a^2 - b^2. So, let's substitute c^2:= a^2 cos² θ - 2 a c cos θ + (a^2 - b^2) + b^2 sin² θ.Simplify:= a^2 cos² θ - 2 a c cos θ + a^2 - b^2 + b^2 sin² θ.Combine like terms:= a^2 (cos² θ + 1) - 2 a c cos θ - b^2 (1 - sin² θ).But 1 - sin² θ = cos² θ, so:= a^2 (cos² θ + 1) - 2 a c cos θ - b^2 cos² θ.Factor cos² θ:= a^2 (1 + cos² θ) - 2 a c cos θ - b^2 cos² θ.Hmm, this is getting complicated. Maybe there's a better way.Wait, perhaps I can express everything in terms of a and c, since b^2 = a^2 - c^2.So, substitute b^2 = a^2 - c^2:= a^2 (cos² θ + 1) - 2 a c cos θ - (a^2 - c^2) cos² θ.Expand:= a^2 cos² θ + a^2 - 2 a c cos θ - a^2 cos² θ + c^2 cos² θ.Simplify:a^2 cos² θ - a^2 cos² θ cancels out.Left with:a^2 - 2 a c cos θ + c^2 cos² θ.Factor:= a^2 - 2 a c cos θ + c^2 cos² θ.Notice that this is equal to (a - c cos θ)^2.Yes, because (a - c cos θ)^2 = a^2 - 2 a c cos θ + c^2 cos² θ.So, the numerator inside the brackets is (a - c cos θ)^2.Therefore, PQ^2 = a^2 (a - c cos θ)^2 / (a - c cos θ)^2 = a^2.Thus, PQ = sqrt(a^2) = a.Since OA = a, we have PQ = OA = OB.So, the statement is proven.

💡Okay, so I've got this piecewise function g(x) defined as:[g(x) =begin{cases}15x + 25 & text{if } x < 0, 3x - 9 & text{if } x geq 0.end{cases}]And I need to find all values of x such that g(x) equals 6, then sum those x values. Hmm, okay. Let me break this down step by step.First, since it's a piecewise function, I should consider both cases separately: when x is less than 0 and when x is greater than or equal to 0. That way, I can solve for x in each scenario and then check if the solutions make sense within their respective domains.Starting with the first case: when x is less than 0, g(x) is defined as 15x + 25. So, I need to set up the equation:15x + 25 = 6Alright, let's solve for x. Subtract 25 from both sides:15x = 6 - 2515x = -19Now, divide both sides by 15:x = -19/15Hmm, let me check if this solution is valid for the domain x < 0. Well, -19/15 is approximately -1.2667, which is indeed less than 0. So, that's a valid solution. Okay, so x = -19/15 is one solution.Now, moving on to the second case: when x is greater than or equal to 0, g(x) is defined as 3x - 9. So, I'll set up the equation:3x - 9 = 6Let's solve for x. Add 9 to both sides:3x = 6 + 93x = 15Now, divide both sides by 3:x = 5Alright, let's check if this solution is valid for the domain x ≥ 0. Well, 5 is definitely greater than 0, so that works. So, x = 5 is another valid solution.Now, the problem asks for the sum of all possible values of x. So, I need to add these two solutions together:-19/15 + 5Hmm, to add these, I should express 5 as a fraction with the same denominator as -19/15. Since 15 is the denominator, 5 can be written as 75/15.So, now:-19/15 + 75/15 = ( -19 + 75 ) / 15 = 56/15So, the sum of all possible values of x is 56/15.Wait, let me double-check my calculations to make sure I didn't make any mistakes.For the first case:15x + 25 = 615x = 6 - 2515x = -19x = -19/15Yep, that's correct.For the second case:3x - 9 = 63x = 15x = 5That's also correct.Adding them together:-19/15 + 5 = -19/15 + 75/15 = 56/15Looks good. So, I think I've got it right.

💡Alright, so I have this problem here: "The tax on a commodity is diminished by 10% and its consumption increases by 10%. Find the effects on revenue." Okay, let's break this down step by step.First, I need to understand what's being asked. It's about how changes in tax rates and consumption affect overall revenue. Revenue, in this context, would be the total money generated from the tax on that commodity. So, if the tax rate goes down but more of the commodity is consumed, does the total revenue increase, decrease, or stay the same?Let me start by defining some variables to make this clearer. Let's say the original tax rate is T, and the original consumption is C. So, the original revenue (R) would be R = T * C. That makes sense because revenue is tax rate multiplied by the amount consumed.Now, the tax is diminished by 10%. That means the new tax rate (let's call it T_new) is 10% less than the original tax rate. To calculate that, I can subtract 10% of T from T itself. So, T_new = T - (10/100)*T. Simplifying that, T_new = T - 0.10*T = 0.90*T. Okay, so the new tax rate is 90% of the original tax rate.Next, the consumption increases by 10%. That means the new consumption (let's call it C_new) is 10% more than the original consumption. To find that, I can add 10% of C to C itself. So, C_new = C + (10/100)*C. Simplifying that, C_new = C + 0.10*C = 1.10*C. Got it, so the new consumption is 110% of the original consumption.Now, to find the new revenue (R_new), I need to multiply the new tax rate by the new consumption. So, R_new = T_new * C_new. Substituting the values I found earlier, R_new = 0.90*T * 1.10*C. Multiplying those together, R_new = 0.99*T*C.Wait a minute, 0.99*T*C is the same as 99% of the original revenue (since R = T*C). So, R_new = 0.99*R. That means the new revenue is 99% of the original revenue, which is a decrease of 1%.But let me double-check that to make sure I didn't make a mistake. If the tax rate decreases by 10%, and consumption increases by 10%, the percentage change in revenue isn't just the sum of these two percentages because they're multiplicative, not additive. So, it's not a 10% decrease plus a 10% increase equals no change. Instead, it's a 10% decrease multiplied by a 10% increase.Mathematically, a 10% decrease is a factor of 0.90, and a 10% increase is a factor of 1.10. Multiplying these together: 0.90 * 1.10 = 0.99, which is a 1% decrease. So, yes, the revenue decreases by 1%.Hmm, but intuitively, if you lower the tax rate, people might buy more of the commodity, which could offset the loss in tax revenue. But in this case, the increase in consumption isn't enough to fully compensate for the decrease in tax rate. So, overall, there's a slight decrease in revenue.Let me think of an example to make this more concrete. Suppose the original tax rate is 10 per unit, and the original consumption is 100 units. So, the original revenue would be 10 * 100 = 1000.If the tax rate is decreased by 10%, the new tax rate is 10 - (10 * 0.10) = 9 per unit. If consumption increases by 10%, the new consumption is 100 + (100 * 0.10) = 110 units. Therefore, the new revenue is 9 * 110 = 990.Comparing the new revenue (990) to the original revenue (1000), we see that the revenue has decreased by 10, which is a 1% decrease. So, this example confirms my earlier calculation.But wait, what if the percentage changes were different? For instance, if the tax rate decreased by 20% and consumption increased by 20%, would the revenue still decrease? Let's see: 0.80 * 1.20 = 0.96, which is a 4% decrease. So, even with larger percentages, the revenue still decreases, but by a larger amount.This makes sense because when you have a percentage decrease and a percentage increase, the overall effect depends on the relative sizes of these percentages. In this case, a 10% decrease and a 10% increase result in a net decrease because the decrease is applied to the original tax rate, and the increase is applied to the original consumption. Since they're multiplicative, the net effect is less than the sum of the individual percentage changes.Another way to think about it is using logarithms. The percentage change in revenue can be approximated by the sum of the percentage changes in tax rate and consumption. So, percentage change in revenue ≈ percentage change in tax rate + percentage change in consumption.In this case, percentage change in revenue ≈ -10% + 10% = 0%. But this is just an approximation and doesn't account for the fact that the changes are multiplicative, not additive. The exact calculation shows a 1% decrease, which is slightly different from the approximation.This discrepancy arises because the approximation assumes that the percentage changes are small and that the second-order terms (like the product of the two percentage changes) can be neglected. However, in reality, even with 10% changes, the second-order term (-10% * 10% = -1%) can't be completely ignored, leading to the exact 1% decrease in revenue.So, in summary, when the tax rate decreases by 10% and consumption increases by 10%, the overall effect on revenue is a 1% decrease. This is because the multiplicative effects of the two changes result in a net decrease, even though the percentage changes are equal in magnitude but opposite in direction.I think I've covered all the bases here. I've defined the variables, calculated the new revenue, checked my work with an example, considered the intuition behind the result, and even looked into why the approximation might differ slightly from the exact calculation. This gives me confidence that the conclusion is correct.

💡Alright, so I have this geometry problem here, and I need to prove that EC is perpendicular to AD in triangle ABC. Let me try to visualize the problem first. We have triangle ABC where AB is longer than AC. AM is the median, so M must be the midpoint of BC. AD is the angle bisector of angle A, so it splits angle BAC into two equal angles. Point E is on the median AM such that ED is parallel to AC. I need to show that EC is perpendicular to AD.Hmm, okay. Let me draw this out mentally. Triangle ABC with AB > AC. So, point A is at the top, B is to the left, and C is to the right. AM is the median, so M is the midpoint of BC. AD is the angle bisector, so it goes from A to some point D on BC. Wait, no, actually, in the problem statement, it's not specified where D is. It just says AD is the angle bisector of angle A. So, D must be on BC because angle bisectors in triangles go from a vertex to the opposite side.So, D is on BC, and E is on AM such that ED is parallel to AC. I need to show that EC is perpendicular to AD. Let me think about the properties I can use here. Since ED is parallel to AC, maybe I can use some properties of similar triangles or something related to parallel lines. Also, since AM is a median, M is the midpoint, so BM equals MC. Maybe I can use coordinate geometry here. Let me assign coordinates to the points to make it easier. Let me place point A at (0, 0), point B at (2b, 0), and point C at (2c, 0). Wait, no, that would make AB and AC both on the x-axis, which might not help. Maybe I should place point A at (0, 0), point B at (2b, 0), and point C at (0, 2c) so that ABC is a right triangle? But the problem doesn't specify that ABC is a right triangle, just that AB > AC.Alternatively, maybe I can use vectors or coordinate geometry without assuming specific coordinates. Let me try coordinate geometry. Let me set point A at (0, 0), point B at (2b, 0), and point C at (2c, 2d). Then, the midpoint M of BC would be at ((2b + 2c)/2, (0 + 2d)/2) = (b + c, d). AD is the angle bisector of angle A. The angle bisector theorem tells us that BD/DC = AB/AC. Since AB > AC, BD > DC. Let me compute AB and AC. AB is the distance from A to B, which is sqrt((2b)^2 + 0^2) = 2b. AC is the distance from A to C, which is sqrt((2c)^2 + (2d)^2) = 2*sqrt(c^2 + d^2). So, BD/DC = AB/AC = (2b)/(2*sqrt(c^2 + d^2)) = b / sqrt(c^2 + d^2).Therefore, BD = (b / (b + sqrt(c^2 + d^2))) * BC. Wait, BC is the length from B to C, which is sqrt((2c - 2b)^2 + (2d - 0)^2) = sqrt(4(c - b)^2 + 4d^2) = 2*sqrt((c - b)^2 + d^2). So, BD = (b / (b + sqrt(c^2 + d^2))) * 2*sqrt((c - b)^2 + d^2). Hmm, this is getting complicated.Maybe instead of coordinate geometry, I can use vector methods. Let me denote vectors for points B and C as vectors b and c respectively. Then, the median AM would be the vector (b + c)/2. The angle bisector AD can be expressed as a vector proportional to (b + c) scaled by some factor. Wait, no, the angle bisector divides BC in the ratio of AB to AC. So, if AB = |b| and AC = |c|, then BD/DC = |b| / |c|. Therefore, point D can be expressed as (|c|*b + |b|*c) / (|b| + |c|). Hmm, that might be useful.Now, point E is on AM such that ED is parallel to AC. Since ED is parallel to AC, the vector ED is a scalar multiple of vector AC. Let me denote vector AC as c. So, vector ED = k*c for some scalar k. Since E is on AM, which is the median, E can be expressed as a point along the line from A to M. So, E = A + t*(M - A) = t*M, since A is at the origin. Wait, no, if A is at the origin, then M is (b + c)/2, so E = t*(b + c)/2 for some t between 0 and 1.But we also have that vector ED = k*c. Vector ED is D - E. So, D - E = k*c. Therefore, E = D - k*c. But E is also equal to t*(b + c)/2. So, t*(b + c)/2 = D - k*c. Let me express D in terms of b and c. From earlier, D = (|c|*b + |b|*c) / (|b| + |c|). So, substituting D into the equation:t*(b + c)/2 = (|c|*b + |b|*c)/( |b| + |c| ) - k*cLet me rearrange this:t*(b + c)/2 + k*c = (|c|*b + |b|*c)/( |b| + |c| )Let me factor out c on the left side:t*(b + c)/2 + k*c = t*b/2 + (t/2 + k)*cSo, equating coefficients:For vector b: t/2 = |c| / (|b| + |c| )For vector c: (t/2 + k) = |b| / (|b| + |c| )From the first equation:t/2 = |c| / (|b| + |c| )So, t = 2|c| / (|b| + |c| )From the second equation:t/2 + k = |b| / (|b| + |c| )Substituting t:(2|c| / (|b| + |c| )) / 2 + k = |b| / (|b| + |c| )Simplify:|c| / (|b| + |c| ) + k = |b| / (|b| + |c| )Therefore, k = |b| / (|b| + |c| ) - |c| / (|b| + |c| ) = (|b| - |c| ) / (|b| + |c| )So, now we have E expressed as t*(b + c)/2, where t = 2|c| / (|b| + |c| ). So, E = (2|c| / (|b| + |c| )) * (b + c)/2 = |c|*(b + c) / (|b| + |c| )So, E = |c|*(b + c) / (|b| + |c| )Now, we need to show that EC is perpendicular to AD. First, let's find vector EC. Vector EC = C - E. Since C is vector c, EC = c - E = c - |c|*(b + c)/( |b| + |c| ) = c*(1 - |c|/( |b| + |c| )) - |c|*b/( |b| + |c| )Simplify:EC = c*( (|b| + |c| - |c| ) / (|b| + |c| ) ) - |c|*b/( |b| + |c| ) = c*|b|/( |b| + |c| ) - |c|*b/( |b| + |c| )So, EC = (|b|c - |c|b ) / (|b| + |c| )Now, vector AD is D - A = D, since A is at the origin. D is (|c|b + |b|c ) / (|b| + |c| )So, AD = (|c|b + |b|c ) / (|b| + |c| )We need to show that EC is perpendicular to AD, which means their dot product is zero.Compute EC · AD:[ (|b|c - |c|b ) / (|b| + |c| ) ] · [ (|c|b + |b|c ) / (|b| + |c| ) ] = [ (|b|c - |c|b ) · (|c|b + |b|c ) ] / (|b| + |c| )^2Let me compute the numerator:(|b|c - |c|b ) · (|c|b + |b|c ) = |b||c| (c · b ) + |b|^2 (c · c ) - |c|^2 (b · b ) - |b||c| (b · c )= |b||c| (c · b ) + |b|^2 |c|^2 - |c|^2 |b|^2 - |b||c| (b · c )Wait, hold on. Let me compute term by term:First term: |b|c · |c|b = |b||c| (c · b )Second term: |b|c · |b|c = |b|^2 (c · c ) = |b|^2 |c|^2Third term: -|c|b · |c|b = -|c|^2 (b · b ) = -|c|^2 |b|^2Fourth term: -|c|b · |b|c = -|b||c| (b · c )So, adding all terms:|b||c| (c · b ) + |b|^2 |c|^2 - |c|^2 |b|^2 - |b||c| (b · c )Notice that the first and fourth terms cancel each other: |b||c| (c · b ) - |b||c| (b · c ) = 0Similarly, the second and third terms cancel each other: |b|^2 |c|^2 - |c|^2 |b|^2 = 0So, the entire numerator is zero. Therefore, EC · AD = 0 / (|b| + |c| )^2 = 0Hence, EC is perpendicular to AD.Wait, that seems too straightforward. Did I make a mistake somewhere? Let me double-check.I expressed vectors in terms of b and c, found E in terms of b and c, computed EC and AD, then took their dot product. The dot product turned out to be zero because the terms canceled out. Hmm, seems correct. Maybe I should try a specific example to verify. Let me take specific coordinates.Let me set point A at (0, 0), point B at (2, 0), and point C at (0, 2). So, AB = 2, AC = 2√2, so AB < AC, but the problem states AB > AC. Let me adjust. Let me set point B at (3, 0) and point C at (0, 1). Then AB = 3, AC = √(0^2 + 1^2) = 1, so AB > AC.Compute M, the midpoint of BC: ((3 + 0)/2, (0 + 1)/2) = (1.5, 0.5)AD is the angle bisector. Using the angle bisector theorem, BD/DC = AB/AC = 3/1 = 3. So, BD = 3*DC. Since BC is from (3,0) to (0,1), its length is √(9 + 1) = √10. So, BD = 3√10 /4, DC = √10 /4.Coordinates of D: Since BD/DC = 3, D divides BC in the ratio 3:1. So, using section formula, D = ( (3*0 + 1*3)/4, (3*1 + 1*0)/4 ) = (3/4, 3/4)So, D is at (0.75, 0.75)Now, E is on AM such that ED is parallel to AC. AM is from A(0,0) to M(1.5, 0.5). Let me parametrize AM. Any point on AM can be written as (1.5t, 0.5t) for t between 0 and 1.Point E is (1.5t, 0.5t). We need ED parallel to AC. Vector ED = D - E = (0.75 - 1.5t, 0.75 - 0.5t). Vector AC = C - A = (0,1). So, ED should be a scalar multiple of AC. That is, (0.75 - 1.5t, 0.75 - 0.5t) = k*(0,1) for some k.Therefore, 0.75 - 1.5t = 0, and 0.75 - 0.5t = k. From the first equation: 0.75 = 1.5t => t = 0.5.So, t = 0.5, so E is (1.5*0.5, 0.5*0.5) = (0.75, 0.25)Now, compute EC and AD.Point E is (0.75, 0.25), point C is (0,1). So, vector EC = C - E = (-0.75, 0.75)Vector AD: Point D is (0.75, 0.75), so vector AD = D - A = (0.75, 0.75)Check if EC is perpendicular to AD. Compute their dot product: (-0.75)(0.75) + (0.75)(0.75) = (-0.5625) + (0.5625) = 0Yes, their dot product is zero, so EC is perpendicular to AD.So, in this specific case, it holds true. Therefore, the proof seems correct.**Final Answer**boxed{EC perp AD}