2202

Prove $R=\sqrt{AP\cdot AQ}$.

2011

The red lines are the perpendicular bisectors of the sides of the triangle. Prove $a+b+c+r = 2R$.

2008

Prove $a+b+c = 2(r+R)$.

2000

Prove radii of the red circles are $\dfrac{Rh_a}{a+h_a}$, $\dfrac{Rh_b}{b+h_b}$, and $\dfrac{Rh_c}{c+h_c}$.

1998

Prove $R = a + b + c$.

1929

Prove the distance between the circumcenter and an excenter is $d = \sqrt{R^2 + 2Rr}$

1889

The red circles are congruent.Prove $s = \dfrac{rR}{2r+R}$.

1588

Prove the radius of the red circle is double the radius of the blue circle.

1583

Prove the radius of the blue circle is double the radius of the red circle.

1462

Prove $a+b+c=r+4R$.

1281

Prove $R=\dfrac{ab}{2h}$.

898

Given $a,b,c$ is an arithmetic progression. Prove $ac=6rR$

701

Prove the ratio of area of the yellow triangle to the area of the red triangle is $r:2R$.

559

Prove $\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=\dfrac{2}{R}$.

383

Prove $a+b+c=R+r$.