Explanation

Let's solve it using complex number geometry.

Now, let the coordinates of triangle be $z_1, z_2, z_3$ and the side lengths be $a, b, c$ and semi-perimeter be $s$

Now, let coordinates of contact triangle be $z_1', z_2', z_3'$

Now, $z_1'$ divides side joining $z_2$ and $z_3$ in ratio $s-b:s-c$ .

So,

$z_1' = \frac{z_2(s-c) + z_3(s-b)}{2s - b - c}$

Similarly

$z_2' = \frac{z_1(s-c) + z_3(s-a)}{2s - a - c}$

$z_3' = \frac{z_2(s-a) + z_1(s-b)}{2s - a - b}$

So, now we have recurrence relation between coordinates of triangle and coordinates of contact triangle, let us write the code to find the coordinates of the points of triangle when all three vertices coincide

For above question, we have $z_1 = 0 + 0i, z_2 = 14 + 0i, z_3 = 5+12i$

Code

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15

z1= 0+0j
z2 = 14+0j
z3 = 5+12j

i = 0

while i <= 54:
    a, b, c = abs(z2 - z3), abs(z3-z1), abs(z1-z2) #lengths of sides
    s = (a+b+c)/2 #semi-perimeter

    z1, z2, z3 = ((s-c)*z2+ (s-b)*z3)/(2*s-b-c), ((s-a)*z3+ (s-c)*z1)/(2*s-a-c), ((s-a)*z2 + (s-b)*z1)/(2*s-b-a) #reassigning vertices to that of contact triangle

    i += 1

print(z1, z2, z3)

Output

1
2
3
4
5

Output:

(5.6715323176778805+3.949803841058493j)
(5.6715323176778805+3.949803841058493j)
(5.6715323176778805+3.949803841058493j)

Answer

So, $P = 5.67153+3.94980i, PM = \text{Im}(P) = 3.94980$

$\color{#3D99F6}{\boxed{\lfloor 10^4 × PM \rfloor = 39498}}$