While substituting for a colleague in Modern Geometry, we were reviewing several theorems regarding circles and doing some constructions with compass and ruler. At one point I presented a theorem on the fact that a tangent line to a circle is perpendicular to the line passing through the center of the circle and the point of the tangent. We also demonstrated that the converse is true: any line that passes through a point on a circle that is perpendicular to the through that point and the center must be a tangent.
We then went though how to construct a tangent to a circle through some external point.
1. Draw the line segment from the center of the circle to the external point.
2. Find the perpendicular bisector of this segment
3. Draw the circle whose center is the the midpoint and passes through the center of the circle and the external point.
4. Draw the line(s) from the external point to the intersection of the two circles.
The line in 4. is tangent. At the time we went through this, I couldn’t see why this was tangent. Shortly afterward it hit me. It’s simply because of Thales Theorem which says that if you have a triangle whose vertices are on a circle and one side is a diameter, then the angle opposite the diameter is right. So by the theorem mentioned above the line(s) is 4. is tangent.
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