×

# 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 $$\triangle ABC$$.

Note by Cody Johnson
3 years, 10 months ago

Sort by:

Is it 98?

- 3 years, 10 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.

- 3 years, 10 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

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

- 3 years, 10 months ago

Correct! 3 points for you!

- 3 years, 10 months ago

Thanks!

- 3 years, 10 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.

- 3 years, 10 months ago

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

- 3 years, 10 months ago

You're welcome. Also, tags? ;)

- 3 years, 10 months ago