(Ahmed Faiz) Find The Radius of The A-Mixtilinear Circle in Terms of AB, AC and Angle A
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Mohammad(Ahmed Faiz) Find The Radius of The A-Mixtilinear Circle in Terms of AB, AC and Angle A
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Mohammad1298. (Ahmed Faiz) Find the radius of the A-mixtilinear circle in terms of AB, AC and angle ∠A.
Lemma
The points of contact of the A-mixtilinear incircle with the sides AB and AC are found by intersecting these
sides with the line through the incenter I perpendicular to the angle bisector AI.
Solution
circle (O,ρ’) tangent to AB, AC at D,E respectively and also tangent to arc BC
I incircle of ∆ ABC, circle (I, ρ) tangent to AB, AC at C’,B’ respectively and also tangent to BC
a+b+c
(AB)=c, (AC)=b, (BC)=a , (AC’)=(AB’)=x, (AD)=(AD’)=x’, (AI)=y, s =
2
(AC’)=(AB’)=
a + b + c 2a b + c − a
x =s−a = − = (1)
2 2 2
∆ ABC: cosine law
a 2 = b 2 + c 2 − 2cb ⋅ cos w ( 2 )
w ρ w
∆ AIC’: tan = ⇒ ρ = x tan ( 3)
2 x 2
w x x
∆ AIC’: cos = ⇒y=
w
( 4)
2 y cos
2
∆ AID:
w y y ( 4) x
cos = ⇒ x' = =
2 x' w w
cos cos 2
2 2
w ρ ' ∆ AIC ρ
∆ AOD: tan= =
2 x' x
w w
( 3),( 5 ) x ⋅ x tan
x sin (1),( 2)
x'ρ 2 2 ⇒
⇒= ρ' = =
x w w
x cos 2 cos3
2 2
w
sin
b + c − b 2 + c 2 − 2cb ⋅ cos w 2
ρ'=
2 w
cos3
2