Given an angleBAC, and a pointDinside it, construct a circle passing throughDand tangent to the sides of the angle.

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- Sep 19th 2010, 02:18 PMMATNTRNGGeometry Construction... circles
Given an angle

*BAC*, and a point*D*inside it, construct a circle passing through*D*and tangent to the sides of the angle.

- Sep 19th 2010, 05:17 PMArchie Meade
Hi MATNTRNG,

First, construct the bisector of the angle.

This is the blue line in the attachment.

Then draw a line from the point of intersection of the 2 given lines to the given point (the red one).

Next, draw any circle with centre on the bisector (the purple one).

Then draw the green line from the centre of this circle to the point where the circle intersects the red line.

A parallel line from the given point to the angle bisector locates the centre of

one of the required circles.

It is simply an enlargement.

To get the other one, join the centre of the purple circle to the 2nd point

of contact with the red line and draw a parallel line from

the given point to the angle bisector. - Sep 19th 2010, 05:55 PMMATNTRNG
Why does this work though? What is the proof behind it?

- Sep 19th 2010, 06:22 PMArchie Meade
If you draw numerous other purple circles of different sizes,

you will see that the lines from their centres to the red line point of contact are all parallel.

This is because if you extend the given lines out to infinity,

when you draw the purple circle, it can be any size at all !!

Imagine that you are a distance from the drawing....

then you can magnify the drawing such that the purple circle is too small to see!

Now roll the circle to the right and keep magnifying it.

When it becomes visible, it will appear just as it had before.