SOLUTION: Please help me solve this problem Equilateral triangle ABC has side lengths 1. Circle O is tangent to sides AB and BC, and is tangent to the perpendicular bisector of BC at M. I

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Question 459746: Please help me solve this problem
Equilateral triangle ABC has side lengths 1. Circle O is tangent to sides AB and BC, and is tangent to the perpendicular bisector of BC at M. If the cevian PC passes through M, compute the area of triangle APC.
This is a tough problem, your help would be extremely appreciated.
Thank you!
Answer by richard1234(7193) (Show Source):
You can put this solution on YOUR website!

First, we can find the values for AM and MD using basic properties of inscribed circles within triangles. If we let AM = x, MD = y, and Q,R be the intersections of the circle and sides BD, BA respectively, we can say that QD = y, BQ = z, etc. and set up a system of three equations in terms of x, y, and z (I won't go too in-detail with this). Solving the system yields and .

Next, extend PC through P to point S such that angle SBC is a right angle:

Here, we conclude that triangles CMD and CSB are similar (due to the fact that SB, MD are parallel), and they have a 1:2 ratio. Since we have found that , then SB is twice that, or .

Now we note that triangles SBP and MAP are also similar, with the ratio unknown. However, we have found SB and MA (two corresponding sides) so we can find AP:PB. We have

We simplify the fraction and replace BP with 1 - AP, since BP + AP = 1.

Solving for AP yields

.

Now we can find the area of triangle APC. This is because the ratio of AP:AB is the ratio of [APC]:[ABC] ([x] denotes area). Here, the area of triangle APC is AP multiplied by the area of ABC (which is easy to find).

The area of equilateral triangle ABC is

.

Hence, the area of triangle APC is

.

You might want to check my arithmetic and algebraic manipulation as I do make silly mistakes once in a while. :)