perplexus.info :: Geometry : Sum of Quad. Radii

Sum of Quad. Radii (Posted on 2008-04-02)

Let ABCD be a cyclic quadrilateral. Let r_A, r_B, r_C, and r_D be the inradii of triangles BCD, CDA, DAB, and ABC respectively.

Prove that r_A + r_C = r_B + r_D.

Submitted by Bractals

Rating: 2.3333 (3 votes)

Solution: (Hide)

If T denotes triangle XYZ, the problem Sum of Radii can be stated as
r_T + R_T = T_X + T_Y + T_Z
For our problem, we have
r_A + R_A = A_B + A_C + A_D r_B + R_B = B_C + B_D + B_A r_C + R_C = C_D + C_A + C_B r_D + R_D = D_A + D_B + D_C
Therefore,
r_A + r_C = r_B + r_D
follows from the fact that
R_A = R_B = R_C = R_D, A_B = B_A, B_C = C_B, C_D = D_C, D_A = A_D, A_C = -C_A, and B_D = -D_B.

Comments: ( You must be logged in to post comments.)

	Subject	Author	Date
	Radii thriller!	Chesca Ciprian	2008-04-03 11:56:59

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