Inscribed ellipse in a pentagon

The idea of ​​the solution is described here .

Python was used to find a solution.

The semi-major and semi-minor axes of the ellipse are equal

$M=\frac{\sqrt{{4149580736921600}- {125240115200 \sqrt{422147689}}}}{25024000}$

$m=\frac{\sqrt{{125240115200 \sqrt{422147689}}+ {4149580736921600}}}{25024000}$

$trunc(100(M+m)) =486$

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73

#
#Solution for problem
#https://brilliant.org/problems/inscribed-ellipse-in-a-pentagon/
#
from math import isclose
from sympy import *

a,b,c,d,h=symbols('a,b,c,d,h')
A=Point(-1,0)
B=Point(3,0)
C=Point(5,3)
D=Point(1,5)
E=Point(-2,1)
print(A,B,C,D,E)
def PE(A,B,C,D,E):
   AD = Line(A, D) 
   EB = Line(E, B)
   H = EB.intersection(AD)[0]
   CH = Line(C, H)
   EA = Line(E, A)
   J = CH.intersection(EA)[0]
   return(J)

def eq_ellipse(x,y):
  return(x*x*a+x*y*b+y*y*c+d*x+h*y)

Q1=PE(A,B,C,D,E)
Q2=PE(B,C,D,E,A)
Q3=PE(C,D,E,A,B)
Q4=PE(D,E,A,B,C)
Q5=PE(E,A,B,C,D)
print(Q1,Q2,Q3,Q4,Q5)
(x1,y1)=Q1
(x2,y2)=Q2
(x3,y3)=Q3
(x4,y4)=Q4
(x5,y5)=Q5
print(x1*1.,y1*1.)
print(x2*1.,y2*1.)
print(x3*1.,y3*1.)
print(x4*1.,y4*1.)
print(x5*1.,y5*1.)

#equation= x1*x1*a+x1*y1*b+c*y1*y1+d*x1+h*y1=1
eq1= eq_ellipse(x1,y1)
eq2= eq_ellipse(x2,y2)
eq3= eq_ellipse(x3,y3)
eq4= eq_ellipse(x4,y4)
eq5= eq_ellipse(x5,y5)

answ=solve([eq1-1,eq2-1,eq3-1,eq4-1,eq5-1],(a,b,c,d,h))
print(answ)
print('a=', a.subs(answ).n())
print('b=', b.subs(answ).n())
print('c=', c.subs(answ).n())
print('d=', d.subs(answ).n())
print('h=', h.subs(answ).n())

major=-sqrt(2*(a*h*h+c*d*d-b*d*h-(b*b-4*a*c))*(a+c+sqrt((a-c)**2+b*b)))/(b*b-4*a*c)
minor=-sqrt(2*(a*h*h+c*d*d-b*d*h-(b*b-4*a*c))*(a+c-sqrt((a-c)**2+b*b)))/(b*b-4*a*c)

major=major.subs(answ)
minor=minor.subs(answ)
print('major=', major)
print('minor=', minor)

print(major.n())
print(minor.n())

print(latex(major))
print(latex(minor))

print((major.n()+minor.n())*100.)

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20

Point2D(-1, 0) Point2D(3, 0) Point2D(5, 3) Point2D(1, 5) Point2D(-2, 1)
Point2D(-311/215, 96/215) Point2D(1/31, 0) Point2D(151/35, 69/35) Point2D(293/73, 255/73) Point2D(-761/559, 1035/559)
-1.44651162790698 0.446511627906977
0.0322580645161290 0
4.31428571428571 1.97142857142857
4.01369863013699 3.49315068493151
-1.36135957066190 1.85152057245081
{a: -961, b: 4364/3, c: -24484/9, d: 62, h: 24076/3}
a= -961.000000000000
b= 1454.66666666667
c= -2720.44444444444
d= 62.0000000000000
h= 8025.33333333333
major= 3*sqrt(4149580736921600/9 - 125240115200*sqrt(422147689)/9)/25024000
minor= 3*sqrt(125240115200*sqrt(422147689)/9 + 4149580736921600/9)/25024000
1.58661696929453
3.27656017688469
\frac{3 \sqrt{\frac{4149580736921600}{9} - \frac{125240115200 \sqrt{422147689}}{9}}}{25024000}
\frac{3 \sqrt{\frac{125240115200 \sqrt{422147689}}{9} + \frac{4149580736921600}{9}}}{25024000}
486.317714617922