Two circles cut orthogonally in A and B . A diameter of one of the circles is drawn cutting the other circle in C and D. Prove that BC⋅AD=AC⋅BD.

Jenny Stafford

Jenny Stafford

Open question

2022-08-14

Two circles cut orthogonally in A and B. A diameter of one of the circles is drawn cutting the other circle in C and D. Prove that B C A D = A C B D.

Answer & Explanation

Leroy Cunningham

Leroy Cunningham

Beginner2022-08-15Added 14 answers

Step 1

O A C = A D C by Tangent-secant theorem, hence O A D O C A A C A D = O C O D ..
Step 2
Similarly B C B D = O C O D , hence  B C B D = A C A D B C A D = A C B D , QED.
grippeb9

grippeb9

Beginner2022-08-16Added 2 answers

Step 1
Orthogonality means that O B B O so OB is tangent to circle with center at O′.
Because of tangent chord property we see that (blue) Δ B C O Δ D B O O B O D = B C B D
Similary (red) we have Δ A C O Δ D A O O A O D = A C A D .
Step 2
Since O A = O B we are done.

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