## Wednesday, September 7, 2011

### Problem 665: Intersecting Circles, Secant, Tangent, Angles, Mind Map

Geometry Problem
Level: Mathematics Education, High School, Honors Geometry, College.

Click the figure below to see the complete problem 665.

#### 4 comments:

1. http://img33.imageshack.us/img33/974/problem665.png
Denote (XYZ)=angle(XYZ)
CO interesect DO’ at F
Note that (ECA)=(ABC)=beta
And (ABD)=(ADE)=alpha
Quadraliteral FCED is cyclic with (CFD)=(ECD)+(EDC)= alpha+beta=(CBD) (see picture)
So 5 points F,C,E,D, B co-cyclic
And ( EBD)=(ECD)= beta ( angles face the same arc DE)
So x= (ABD)-(EBD)= alpha-beta
Peter Tran

2. Join BC, BD
∠CBD = ∠ABD + ∠ABC = α + β
∠CED = 180° − (α + β)
∴B, C, E, D are concyclic
So α = ∠CBE = x + ∠ABC = x + β
∴ x = α − β

3. http://imgsrc.baidu.com/forum/pic/item/38dbb6fd5266d0162ed21d77972bd40734fa35a1.jpg

chinese

4. < CBA = beta and < ABD = alpha hence < CBD + CED = 180

So < EBD = beta = alpha - x hence x = alpha - beta

Sumith Peiris
Moratuwa
Sri Lanka