CMC - Problem 3

Problem 3. (3 points) In $\triangle ABC$, let $D$ be the midpoint of $\overline{BC}$. If $AB=13$, $BC=14$, and $CA=15$, find the maximum value of $(A'B+A'D)(A'B-A'D)+(A'C+A'D)(A'C-A'D)$, where $A'$ can be any point in the plane of $\triangle ABC$.

#Geometry #CMC #Competitions #MathProblem #Math

Easy Math Editor

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Comments

Is it 98?

敬全钟 - 7 years, 6 months ago

Yes, by the Median length theorem aka Apollonius' Theorem(you can call it a special case of Stewarts but the proof is trivial using cosine rule), the expression is equal to $A'B^2+A'C^2-2A'D^2=2BD^2=2*7^2=98$ . This means that the expression stays invariant regardless of the location of $A'$ .

More on the theorem: http://en.wikipedia.org/wiki/Apollonius'_theorem

Sometimes being well-known to these theorems and the structure of some equations can really help.

Xuming Liang - 7 years, 6 months ago

I do in this way. First, I expand the given expression,

$(A'B + A'D)(A'B - A'D) + (A'C + A'D)(A'C - A'D) = (A'B)^{2} + (A'C)^{2} - 2(A'D)^{2} - (1)$

When we connect $A'B, A'D, A'C$ , we will get two new triangles, i.e. $\Delta A'BD$ and $\Delta A'CD$ . So, I apply The Theorem of Stewart,

$7(A'B)^{2} + 7(A'C)^{2} = 14((A'D)^{2} +49)$

Divide 7 on both sides,

$(A'B)^{2} + (A'C)^{2} = 2((A'D)^{2} + 49)$

Expand RHS and subtract $2(A'D)^{2}$ from both sides,

$(A'B)^{2} + (A'C)^{2} - 2(A'D)^{2} = 98$

Comparing the equation above with $(1)$ , we get

$(A'B + A'D)(A'B - A'D) + (A'C + A'D)(A'C - A'D) = \boxed {98}$

敬全钟 - 7 years, 6 months ago

Correct! 3 points for you!

Cody Johnson - 7 years, 6 months ago

@Cody Johnson – Thanks!

敬全钟 - 7 years, 6 months ago

Too sleepy to solve it right now, but a little piece of humble advice on the problem-writing. The way you have defined $A'$ is not logically incorrect but it is a bit confusing and it is not the standard way of writing it. Here is how I would rephrase the problem. Also, "in" is not completely rigorous.

Let $ABC$ be a triangle such that $AB = 13$ , $BC = 14$ , and $CA = 15$ . Let $D$ be the midpoint of $\overline {BC}$ . Let $A'$ be a point coplanar to $\triangle ABC$ . Find the maximum possible value of $\left( A'B + A'D \right) \cdot \left(A'B - A'D\right) + \left(A'C + A'D\right) \cdot \left(A'C - A'D\right).$

Hope you don't take this as criticism or egotistical advice; rather, as suggestions on problem-writing.

Ahaan Rungta - 7 years, 6 months ago

I had that at first but was tentative. Thanks for your criticism, I genuinely appreciate it.

Cody Johnson - 7 years, 6 months ago

You're welcome. Also, tags? ;)

Ahaan Rungta - 7 years, 6 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$