1. ## 4 equations

4 equations, 4 unknowns:

u(r^2 - u^2) / (r^2 + u^2) = 156 [1]
v(r^2 - v^2) / (r^2 + v^2) = 96 [2]
w(r^2 - w^2) / (r^2 + w^2) = 63 [3]
(315uvw + 24336vw + 9216uw + 3969uv) / (2r) = 943488 [4]

Who can solve that mess?

2. ## Re: 4 equations

Did you notice that all the integers are divisible by three? From equation four, we know that r is also divisible by three.

3. ## Re: 4 equations

Originally Posted by zhandele
...From equation four, we know that r is also divisible by three.
Not so. I brute forced this one: r = 520. And u=260, v=104, w=65.

I'm trying to do this by solving instead of brute force.

4. ## Re: 4 equations

Hi Wilmer, can you give us some clue as to background/origin ?

5. ## Re: 4 equations

Thanks Bob. Best I can do diagramwise(!):
Code:
              C

D                     E
U          V
M

W

B                  F                        A
Acute triangle ABC: M is circumcenter.
U, V and W are the incenters of triangles BCM, ACM and ABM respectively:
and DM, EM and FM are the perpendicular heights.

NOT GIVENS: a = BC = 624, b = AC = 960, c = AB = 1008, r = 520 = AM=BM=CM.
NOT GIVENS: u = UM = 260, v = VM = 104, w = WM = 65.
GIVENS: d = DU = 156, e = EV = 96, f = FW = 63.

Work to set up the 4 equations:

a = 2dr / u , b = 2er / b , c = 2fr / w

from triangleBCM: u(r^2 - u^2) / (r^2 + u^2) = d [1]
from triangleACM: v(r^2 - v^2) / (r^2 + v^2) = e [2]
from triangleABM: w(r^2 - w^2) / (r^2 + w^2) = f [3]

area(BCM + ACM + ABM) = areaABC; leads to :
[uvw(d + e + f) + vwd^2 + uwe^2 + uvf^2] / (2r) = def [4]

Inserting the givens gives us:

u(r^2 - u^2) / (r^2 + u^2) = 156 [1]
v(r^2 - v^2) / (r^2 + v^2) = 96 [2]
w(r^2 - w^2) / (r^2 + w^2) = 63 [3]
(315uvw + 24336vw + 9216uw + 3969uv) / (2r) = 943488 [4]

I'm simply curious as to the possibility of solving these 4 simultaneous equations.