...FI) E and ABC are equal, (Cor. Theo. 32.) QED THEOREM XXXVI. If two equal triangles as ABC, AD B, be on the same base AB, and on the same side of it, the line DC which joins their vertices, is parallel to the base A B. For if D С is not parallel to...

...B and C are equal. COR. — Hence every equiangular triangle is also equilateral. PROP. VII. THEOR. On the same base (AB), and on the same side of it, there cannot be two triangles having their conterminous sides (AC and AD, BC and BD) at both extremities...

...likewise those which are terminated in the other extremity, equal to one another. If it be possible, upon the same base AB, and on the same side of it, let ACB, ADB be two triangles, which have their sides CA, DA terminated in A, one extremity of the base,...

...draw EG perpendicular to DB and meeting DC produced in G. Then CEG is an equilateral triangle. 69. On the same base AB, and on the same side of it, let...angle ABD not a right angle ; then the triangle ABC may be shewn to oe greater than the triangle ABD whether the angle ABD be acute or obtuse. 70. Let...

...draw EG perpendicular to DB and meeting DC produced in G. Then CEG is an equilateral triangle. 69. On the same base AB, and on the same side of it, let...angle ABD not a right angle; then the triangle ABC may be shewn to oe greater than the triangle ABD whether the angle ABD be acute or obtuse. 70. Let...

...those sides which terminate in the other end of the base are not equal. Let the triangles ACB and ADB, on the same base AB, and on the same side of it, have the sides AC and AD equal; then the sides вc and BD are not equal. CASE 1. When the vertex of...

...BD by hyp8 : which is impossible. Hence in every case it has been shewn impossible for there to be on the same base AB, and on the same side of it, two triangles ACB, ADB, having their sides AC, AD terminated in one extremity A of the base equal,...

...ways, and the figure may be transformed into a triangle, by two constructions similar to that employetl for a figure of four sides. If the figure consists...be a triangle whose vertical angle is A, and whose base BC is bisected in D ; let any line EDG be drawn through D, meeting AC the greater side in G and...

...bisection of the base parallel to the two sides of the triangle. 108. This may be shewn ex absurdo. 1 09. On the same base AB, and on the same side of it, let...constructed, having the side BD equal to BC, the angle ABC aright angle, but the angle ABD not a right angle; then the triangle ABC is greater than ABD, whether...

...AECG the equivalent rectangle. Then AC is greater than EA, &c. 17. This may be shewn by Euc. I 20. 18. On the same base AB, and on the same side of it, let...than ABD, whether the angle ABD be acute or obtuse. 19. Make an isosceles triangle, having its vertical angle equal to the given angle. Describe a triangle...