## Friday, September 23, 2011

### Problem 672: Internally tangent circles, Chord, Tangent, Geometric Mean

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

Click the figure below to see the complete problem 672.

1. Extend AD to meet circle O again at E
Circles O and O' touch each other at A
So points A, O', O are collinear
OA being a diameter(of circle O'),
ODA is a right angle
So OD bisects the chord ADE in circle O

2. Extend AD to meet circle O at E. Since OA is a diameter of circle O',OD is perpendicular to AD or OD is perpendicular chord AE of circle O. In other words D is the midpt. of AE which means AD*DC = AD*DE = AD^2 (intersecting chords) or a*b = AD^2 or AD is the GM of a & b.
Ajit : ajitathle@gmail.com

3. Typo Correction:
Ajit

4. Chinese version
http://imgsrc.baidu.com/forum/pic/item/500fd9f9d72a6059e27a5d452834349b033bba03.jpg

5. Let AD extended meet circle O at P

< ODA = 90 so PD = DA = x

Hence easily in circle O,

PD. DA = a.b i.e. x^2 = a.b

Sumith Peiris
Moratuwa
Sri Lanka