math-geo-01

# Incircle Contact Chord Meets the Base

In triangle $ABC$ we have $AB = 13$, $BC = 14$, and $CA = 15$.

The incircle of triangle $ABC$ is tangent to side $AB$ at $F$ and tangent to side $CA$ at $E$. Line $EF$ is extended until it meets line $BC$ at a point $P$.

Find the length $BP$.

Give your final answer as an integer.

Alright, I have a problem to solve:

**Problem Statement:**
In triangle ABC, AB = 13, BC = 14, and CA = 15. The incircle of triangle ABC is tangent to side AB at F and tangent to side CA at E. Line EF is extended until it meets line BC at a point P. Find the length BP.

### Understanding the Problem
First, I need to draw the triangle ABC with the given side lengths. Points E and F are points of tangency of the incircle with sides CA and AB, respectively. The line EF is extended to meet BC at P, and I need to find the length BP.

### Step 1: Find the Coordinates or Use Trigonometry
I'm not sure if I should use coordinate geometry or stick to pure geometry. Since the problem gives exact side lengths, maybe I can find the area and other properties first.

### Step 2: Find the Area of Triangle ABC
I recall Heron's formula can be used here. First, calculate the semi-perimeter (s):
s = (a + b + c)/2 = (13 + 14 + 15)/2 = 42/2 = 21.

Now, area (A) = √[s(s - a)(s - b)(s - c)] = √[21 * (21 - 13) * (21 - 14) * (21 - 15)] = √[21 * 8 * 7 * 6] = √[21 * 8 * 7 * 6].

Let me simplify this:
21 * 7 = 147
8 * 6 = 48
So, A = √[147 * 48] = √[147 * 48] = √[147 * 48].

But 147 = 49 * 3 and 48 = 16 * 3, so:
A = √[49 * 3 * 16 * 3] = √[49 * 16 * 9] = 7 * 4 * 3 = 84.

### Step 3: Find the Inradius
The area of the triangle is also given by A = r * s, where r is the inradius.
So, 84 = r * 21 → r = 84 / 21 = 4.

### Step 4: Find the Points of Tangency E and F
The incircle is tangent to the sides. The lengths of the tangents from the vertices to the points of tangency can be found using the formula:
For any triangle, the length of the tangent from vertex A to the point of tangency on AB is s - a, but I'm not sure.

Actually, the formula is:
- The length of the tangent from A to the point of tangency on AB is s - BC.
- The length of the tangent from A to the point of tangency on AC is also s - BC.

But in this case:
- The point of tangency on AB is F, so AF = s - BC = 21 - 14 = 7.
- The point of tangency on AC is E, so AE = s - BC = 21 - 14 = 7.

But, the point of tangency on BC is D, so BD = s - AC = 21 - 15 = 6, and DC = s - AB = 21 - 13 = 8.

### Verification
Let's verify:
AF + BF = AB → 7 + BF = 13 → BF = 6.
AE + CE = AC → 7 + CE = 15 → CE = 8.
BD + DC = BC → 6 + 8 = 14 → Correct.

### Step 5: Find the Length of EF
Points E and F are points of tangency on AC and AB, respectively. The incircle is tangent to both sides, so the distances from the vertices to the points of tangency are known.

But how to find EF? I recall that the line joining the points of tangency of the incircle with two sides is related to the angle between those sides.

First, I need the angle at A. I can use the cosine rule:
cos(A) = (AB² + AC² - BC²) / (2 * AB * AC) = (13² + 15² - 14²) / (2 * 13 * 15) = (169 + 225 - 196) / 390 = (394 - 196)/390 = 198/390 = 33/65.

Now, the length of EF can be found using the formula for the distance between two points of tangency:
EF = √[AE² + AF² - 2 * AE * AF * cos(A)] = √[7² + 7² - 2 * 7 * 7 * (33/65)] = √[49 + 49 - 98 * (33/65)] = √[98 - (3234/65)].

Let's simplify 98 - (3234/65):
98 = 6370/65 → 6370/65 - 3234/65 = 3136/65.

Thus, EF = √(3136/65) = √(3136)/√65 = 56/√65.

But I need to rationalize the denominator: 56√65 / 65.

### Step 6: Find the Equation of Line EF and Its Extension to P
This seems complex. Maybe I should use coordinate geometry.

### Step 7: Assign Coordinates to Triangle ABC
Let's place point A at the origin (0,0), AB along the x-axis.

- A = (0,0)
- B = (13,0) since AB = 13.

Now, find the coordinates of C. We know AC = 15 and BC = 14.

Let C = (x, y).

Then:
√(x² + y²) = 15 → x² + y² = 225.
√[(x - 13)² + y²] = 14 → (x - 13)² + y² = 196.

Subtract the second equation from the first:
x² + y² - [(x - 13)² + y²] = 225 - 196 → x² - (x² - 26x + 169) = 29 → 26x - 169 = 29 → 26x = 198 → x = 198/26 = 99/13.

Now, y² = 225 - (99/13)² = 225 - 9801/169 = (225*169 - 9801)/169 = (38025 - 9801)/169 = 28224/169 → y = √(28224)/13 = 168/13.

So