By A. N. Whitehead
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Extra resources for The Axioms of Descriptive Geometry (Dover Phoenix Editions)
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For example, we might mark off a segment equal to OA on our straightedge, then maneuver the straightedge so that it passes through E, and the endpoints of the marked segment lie on the circle and the secant. But marking a segment, or even a point, on the straightedge violates Euclid’s conventions. Exercise 50. A circle passes through two fixed points A, B. Let C be one of the points where this circle meets a fixed line perpendicular to AB. Find the locus of the point diametrically opposite to C as the circle varies while passing through A and B.
We are given a right angle AOB and two perpendicular lines through a point P , the first intersecting the sides of the angle in A, B and the second intersecting the same sides in C, D. Show that the perpendiculars from points D, O, C to line OP intercept on AB segments equal to AP , P B, respectively, but situated in the opposite sense. Solution. In Figure t46, we must show that XY = P A and Y Z = P B. Suppose K is the midpoint of segment AD. Now in triangles AP D, AOD, medians KP and KO are equal to half of the hypotenuse (48), so KA = KP = KO = KD.
Lemma 2. A line through the midpoint of one diagonal of a trapezoid, parallel to the bases, bisects the other diagonal of the figure. A C B A o C o B o Figure t35a Solution. Let AA0 and BB0 be the perpendiculars from points A and B to the given line, let C be the midpoint of segment AB, and draw CC0 ⊥ A0 B0 . If points A and B lie on the same side of the given line (Figure t35a), then AA0 B0 B is a trapezoid. Line CC0 is parallel to the bases of the trapezoid, so by Lemma 1, it must bisect A0 B0 .