# Equilateral triangle ABC is inscribed in a circle

1. Nov 29, 2014

### dawo0

1. The problem statement, all variables and given/known data
Can someone help me solve this, and teach me how to solve such problems in future? An equilateral triangle ABC is inscribed in a circle . Point D lies on a shorter arc of a circle BC. Point E is symmetrical the point B relating to the line CD . Prove that the points A, D , E lie on one straight line. If something is unclear just tell me.

2. Relevant equations
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3. The attempt at a solution

I do not know how to go about Himself .
Frame I figure .

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2. Nov 29, 2014

### ehild

Welcome to PF!

Use the Theorem of Inscribed Angles.

3. Nov 29, 2014

### HallsofIvy

Staff Emeritus
Excellent English! Much better than before. Congratulations. This may be more than is necessary but I am more comfortable with "analysis" than "geometry" so I would set up a coordinate system with origin at the center of the circle and (0, 1) at point C. Then the equation of the circle, in that coordinate system, is $x^2+ y^2= 1$. On can show that the line through the point A and the center of the circle is $y= x/2$ so that point A has coordinates where that line intersects the circle. With $y= x/2$ the equation of the circle becomes $x^2+ x^2/4= (5/4)x^2= 1$ so that $x= -\frac{2}{\sqrt{5}}= -\frac{2\sqrt{5}}{5}$ and $y= -\frac{\sqrt{5}}{5}$. Similarly for point B. With that information it should be fairly easy to find the coordinates of points D and E and show that they line on a single line.

4. Nov 29, 2014

### dawo0

I figured all the angles. What next?

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• ###### Kąty.png
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5. Nov 29, 2014

### ehild

Try to determine the angle ADE. (First find the angle of the blue right triangle at D.

6. Nov 29, 2014

### dawo0

How do I determine the angle ADE as it is simple? .
Calculated the angles of the blue triangle.
Help ! "D l

7. Nov 29, 2014

### ehild

You have to prove that ADE is a straight line, that is, the angle ADE is 180°. What are the angles ABC, ADC ADB,?
The red line CD halves the blue triangle, as it is symmetric to the red line. Wjhat are the angles at D?

8. Nov 29, 2014

### dawo0

All I calculated.
I can't write request.

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• ###### 29 listopad 2014 (2).gif
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9. Nov 29, 2014

### ehild

Show what you calculated. What are the angles γ and δ; β and λ? How many degrees? What is the angle σ?

10. Nov 29, 2014

### dawo0

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• ###### circleangles.JPG
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11. Nov 29, 2014

### RUber

You seem to be assuming that the line drawn from A will intersect BC at a 90 degree angle. This is incorrect.
In post #2, ehild said to use the law of inscribed angles. Check that out and see if you can find all the angles in terms of the one variable angle.

12. Nov 30, 2014

### ehild

@dawo0: your drawing does not help to solve the problem. It shows the special case when the point D is at the middle of the arc BC. But you need to prove the statement of the problem for any point D on the arc.
You have drawn ADE as a straight line, but you have to prove that it is a straight line!
Do you understand why are γ=δ=60°and λ=β=60°? You should explain. In general, you should explain your statements when solving a problem.
Something like that: β is one angle of the equilateral triangle, so it is 60°. β and λ are inscribed angles of the circle, belonging to the same arc AC, so λ=β...

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