Elements of Plane Geometry

Cover Elements of Plane Geometry
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The diagonals of a parallelogram mutually bisect each other (Prop. IV. ); hence AB^+AC2=:2EB2+2AE2, and BD2+DC2=. 2BE2+2DE2 (Prop. XXL). Add these equa- tions, and AB2+AC24- BD2+DC2rzi4BE2 +2 AE2+2ED2; or AB2+AC2+BD2+DC2=4BE2+4AE2. But since the square of a line is equal to four times the square of its half, AB2+AC2+BD2+DC2=AD2+BC2.
The foUoicing are Test Examples involving the First and Second JBooks solved or demonstrated : 1. To bisect a parallelogram from a point in 07ie of its sides.
Let A
...BCD be D EC the given paral- lelogram, and P the given point.
On the line DC lay off a segment EC equal AP, and join the points P and E. The line PE bisects the parallelogram.
For, draw the diagonal AC* The two triangles ABC * The student may readily invent other methods of demonstrating this truth by drawing different hues.
64 ELEMENTS OF PLANE GEOMETRY.
and CDA are equal (Prop. III. ). In the two triangles APO and CEO the angles GAP and APO are equal to OCE and CEO, each to each (Prop. XXYH. , Bk.


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