Discussion about math, puzzles, games and fun.   Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ -¹ ² ³ °

You are not logged in.

## #1 2013-04-12 14:12:30

Agnishom
Real Member

Online

### Arcs and Chords

In ΔABC, the perpendicular from vertices A and B on their opposite sides meet (when produced) the circumcircle of the triangle at points D and E respectively.
Prove that arc CD = arc CE

Note: You are not allowed to use trigonometry

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'The whole person changes, why can't a habit?' -Alokananda

## #2 2013-04-12 17:57:20

bob bundy
Moderator

Offline

### Re: Arcs and Chords

Hi Agnishom

For arc CD = arc CE you want angle DAC = angle CBE

See diagram.  Where the altitudes meet the sides I've named those points F and G

So what can you say about triangles AFC and BGC  ?

Bob

You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

## #3 2013-04-12 22:37:38

Agnishom
Real Member

Online

### Re: Arcs and Chords

They have one common angle and one right angle so the other angle must be the same.

thanks but I need a way to say that if angle angle DAC = angle CBE, then CD = CE

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'The whole person changes, why can't a habit?' -Alokananda

## #4 2013-04-12 22:54:39

bob bundy
Moderator

Offline

### Re: Arcs and Chords

Let O be the point at the centre of the circle.

angle DOC = 2 x angle DAC (angle properties of a circle)

similarly angle COE = 2 x angle CBE

OD = OC = OE = radius

So triangles DOC and COE are congruent => straight line DC = straight line CE => arc DC = arc CE

Bob

You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

## #5 2013-04-13 02:50:57

Agnishom
Real Member

Online

### Re: Arcs and Chords

Okay Got it Thanks!

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'The whole person changes, why can't a habit?' -Alokananda