Adopting side aa as the triangle base, the upper vertice describes the ellipse
(x/r_x)^2+(y/r_y)^2=1(xrx)2+(yry)2=1
where
r_x = (a+b+c)/2rx=a+b+c2 and r_y = sqrt(((b+c)/2)^2-(a/2)^2)ry=√(b+c2)2−(a2)2
when y_v = h_0yv=h0 then x_v = (sqrt[a^2 - (b + c)^2 + 4 h_0^2] p_0)/(2 sqrt[a^2 - (b + c)^2])xv=√a2−(b+c)2+4h20p02√a2−(b+c)2. Here p_v={x_v,y_v}pv={xv,yv} are the upper vertice coordinates p_0=a+b+cp0=a+b+c and p=p_0/2p=p02.
The ellipse focuses location are:
f_1 = {-a/2,0}f1={−a2,0} and f_2 = {a/2,0}f2={a2,0}
Now we have the relationships:
1) p (p-a) (p-b) (p-c) = (a^2 h_0^2)/4p(p−a)(p−b)(p−c)=a2h204 Henon´s formula
2) From a + norm(p_v-f_1)+norm(p_v-f_2) = p_0a+∥pv−f1∥+∥pv−f2∥=p0 we have
a + sqrt[h_0^2 + 1/4 (a - (sqrt[a^2 - (b + c)^2 + 4 h_0^2] p_0)/sqrt[
a^2 - (b + c)^2])^2] + sqrt[
h_0^2 + 1/4 (a + (sqrt[a^2 - (b + c)^2 + 4 h_0^2] p_0)/sqrt[
a^2 - (b + c)^2])^2] = p_0a+
⎷h20+14⎛⎜
⎜⎝a−√a2−(b+c)2+4h20p0√a2−(b+c)2⎞⎟
⎟⎠2+
⎷h20+14⎛⎜
⎜⎝a+√a2−(b+c)2+4h20p0√a2−(b+c)2⎞⎟
⎟⎠2=p0
3) a+b+c=p_0a+b+c=p0
Solving 1,2,3 for a,b,ca,b,c gives
(
a = ( p_0^2-4 h_0^2)/(2 p_0),
b= (4 h_0^2 + p_0^2)/(4 p_0),
c= (4 h_0^2 + p_0^2)/(4 p_0)
)(a=p20−4h202p0,b=4h20+p204p0,c=4h20+p204p0)
and substituting h_0=0.173, p_0=0.60h0=0.173,p0=0.60
{a = 0.200237, b = 0.199882, c = 0.199882}{a=0.200237,b=0.199882,c=0.199882}
with an area of 0.01732050.0173205
