# Hadamard's Plane Geometry by Mark Saul

By Mark Saul

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Extra info for Hadamard's Plane Geometry

Example text

Which is the greatest of the three angles (Exercise 7)? Generalize to more than three equal parts. Solution. In Figure t51a, we let AD = DE = EB. Dropping perpendicular OH onto AB from center O of the circle, we see that OE = OD < OA (from the Theorem of 29 about the length of oblique segments). Line OD is a median in triangle OAE, so from the inequality OE < OA, we have (Exercise 7) that DOE > DOA. Similar reaasoning shows that DOE > EOB. Similarly, if chord AB is divided into n equal parts, then the largest central angle will be the one inside of which the perpendicular OJ is located (for odd n), or either of the two angles with OJ on their side (for even n).

Since ABC and A B C are equal by hypothesis with the same orientation, it follows that A BC and ABC are likewise equal with the same orientation. Therefore, C BC and A BA are equal, since they can be obtained by adding C BA to ABC and A BC , respectively. Now BC is parallel to B C , and B C is perpendicular to BC, so C BC is a right angle. Therefore, A BA is also a right angle. It follows that BA is perpendicular to AB, as is B A . In the same way, we can show that A C is perpendicular to AC. Solutions.

This construction, unlike the others, requires the introduction of a whole new ﬁgure—a parallelogram—into the discussion, and thus it is more diﬃcult to think of than the others in this series. On the other hand, there are no exceptional cases. Students can verify this by experimenting with their software sketch. 3◦ . If lines a, b, c contain the angle bisectors of the required triangle ABC, then AC and BC must be symmetric with respect to line b, and AB and CB must be 30 1. SOLUTIONS FOR BOOK I l b C A' O a A A'' B c Figure t38c symmetric with respect to line c (Figure t38c).