Consider a triangle PQR where PQ = 2014, QR = 2015, RP =2016. RM is an altitude of this triangle.
Determine the distance between the points at which the incircles of PRM and QRM touch RM.
Let a = PM, r-a = MQ, h = RM, p = QR, and q = RP.
Applying the Pythagorean theorem to triangles
PMR and QMR gives
h^2 = q^2 - a^2 = p^2 - (r-a)^2
==>
2a = (q^2 + r^2 - p^2)/r
The distance between tangency points is
| a + h - q (r-a) + h - p |
d = | ----------- - --------------- |
| 2 2 |
| p - q - r + 2a |
= | ---------------- |
| 2 |
| p - q - r + (q^2 + r^2 - p^2)/r |
= | --------------------------------- |
| 2 |
= | q(q - r) - p(p - r) |/(2r)
For our problem,
d = | 2016(2016 - 2014) - 2015(2015 - 2014) | / 4028
= 2017/4028 ~= 0.5007447864945382323733862959285
QED
Edited on November 4, 2015, 2:07 pm
|
|
Posted by Bractals
on 2015-11-04 14:05:04 |
Solution
