Elements of geometry and trigonometry . eter^w what is the same, AD-= Hence also, in the second place, the chord AB is a mean pro-portional between the diameter BC and the adjacent segment BD,or, what is the same, AB^ — In like manner, we haveAC2= ; hence AB^ : AC^ : : BD : DC : and com-paring AB^ and AC^, to BC^, we have AB^ : BC^ : ; BD :BC, and AC^ : BC^ : : DC : BC. Those proportions betweenthe squares of the sides compared with each other, or with th«square of the hypothenuse, have already been given in the thirdand fourth corollaries of Prop. XI. PROPOSITION XXIV. THEO
Elements of geometry and trigonometry . eter^w what is the same, AD-= Hence also, in the second place, the chord AB is a mean pro-portional between the diameter BC and the adjacent segment BD,or, what is the same, AB^ — In like manner, we haveAC2= ; hence AB^ : AC^ : : BD : DC : and com-paring AB^ and AC^, to BC^, we have AB^ : BC^ : ; BD :BC, and AC^ : BC^ : : DC : BC. Those proportions betweenthe squares of the sides compared with each other, or with th«square of the hypothenuse, have already been given in the thirdand fourth corollaries of Prop. XI. PROPOSITION XXIV. THEOREM. Two triangles having an angle in each equal, are to each otheras the rectangles of the sides which contain the equal angles. In the two triangles ABC, ADE, let the angle A be equal tothe angle A ; then will the triangle ABC : ADE : : : Draw BE. The trianglesABE, ADE, having the com-mon vertex E, have the samealtitude, and consequently areto each other as their bases(Prop. VI. Cor.) : that is, ABE : ADE : : AB : In like manner, ABC ABE : : AC : AE. Multiply together the corresponding terms of these proportions,omitting tjie common term ABE ; we have ABC : ADE : : BOOK IV 91
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