Equilateral triangle ABC is inscribed in a circle

[dawo0]

Homework Statement

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.

Homework Equations

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The Attempt at a Solution

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

 

[ehild]

Welcome to PF!

Homework Statement

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.

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Use the Theorem of Inscribed Angles.

 

[HallsofIvy]

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.

 

[dawo0]

I figured all the angles. What next?

 

[ehild]

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

 

[dawo0]

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

 

[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?

 

[dawo0]

All I calculated.
I can't write request.

 

[ehild]

All I calculated.
I can't write request.

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

 

[dawo0]

Already pasted.

 

[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.

 

[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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