Q8 LOGICAL DEFICIENCIES IN EUCLIDEAN GEOMETRY15graphical properties of figures, especially those encountered in Sections 1,2, 3.For the sake of clarity we use AB to denote segment AB.1. Let two circles meet in points A, B. Let AC, ĀD be their respectivediameters from A. Prove that B is on line CD.2. From a point P on side BA of ZABC, a perpendicular is dropped ontoside BC (extended if necessary) of the angle. Prove that the foot of theperpendicular is on side BC of ZABC or on the extension of side BCaccording as LABC is acute or obtuse.3. In AABC let AB÷ BC. Prove that the bisector of ZB and the per-pendicular bisector of AC meet in just one point. (That is, they have apoint, but not two points, in common.)4. In AABC let AB÷ BC, and let E be the point of intersection of thebisector of ZB and the perpendicular bisector of AC. Prove that A, B, C,and E lie on a circle. (Note how strongly this supports the conclusionthat E is outside AABC.)5. In AABC let the bisector of ZB meet the circle containing A, B, C in apoint E distinct from B. Prove that BE is a diameter of the circle if andonly if AB = BC.6. In AABC let AB# BC, and let the bisector of ZB meet the perpen-dicular bisector of AC in E. Prove that one of the angles BAE, BCE isacute, and the other obtuse.7. In AABC let AB+ BC, and let the bisector of ZB meet the perpen-dicular bisector of AC in E. From E drop perpendiculars to sides AB, BC(extended if necessary) of the triangle. Prove that one of the feet of theperpendiculars falls on the side to which it was dropped, and the otheron an extension of the side.8. Prove that every triangle has an "interior" altitude (one that meets theside to which it is drawn).9. How many interior altitudes can a triangle have? Justify your answer.10. Criticize the following proof that any right angle is obtuse.BE

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8. Prove that every triangle has an "interior" altitude (one that meets the side to which it is drawn).

Algebra & Trigonometry with Analytic Geometry

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LOGICAL DEFICIENCIES IN EUCLIDEAN GEOMETRY
15
graphical properties of figures, especially those encountered in Sections 1,2, 3.
For the sake of clarity we use AB to denote segment AB.
1. Let two circles meet in points A, B. Let AC, ĀD be their respective
diameters from A. Prove that B is on line CD.
2. From a point P on side BA of ZABC, a perpendicular is dropped onto
side BC (extended if necessary) of the angle. Prove that the foot of the
perpendicular is on side BC of ZABC or on the extension of side BC
according as LABC is acute or obtuse.
3. In AABC let AB÷ BC. Prove that the bisector of ZB and the per-
pendicular bisector of AC meet in just one point. (That is, they have a
point, but not two points, in common.)
4. In AABC let AB÷ BC, and let E be the point of intersection of the
bisector of ZB and the perpendicular bisector of AC. Prove that A, B, C,
and E lie on a circle. (Note how strongly this supports the conclusion
that E is outside AABC.)
5. In AABC let the bisector of ZB meet the circle containing A, B, C in a
point E distinct from B. Prove that BE is a diameter of the circle if and
only if AB = BC.
6. In AABC let AB# BC, and let the bisector of ZB meet the perpen-
dicular bisector of AC in E. Prove that one of the angles BAE, BCE is
acute, and the other obtuse.
7. In AABC let AB+ BC, and let the bisector of ZB meet the perpen-
dicular bisector of AC in E. From E drop perpendiculars to sides AB, BC
(extended if necessary) of the triangle. Prove that one of the feet of the
perpendiculars falls on the side to which it was dropped, and the other
on an extension of the side.
8. Prove that every triangle has an "interior" altitude (one that meets the
side to which it is drawn).
9. How many interior altitudes can a triangle have? Justify your answer.
10. Criticize the following proof that any right angle is obtuse.
B
E

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