Euclid+Book+I

Proposition 37
//Triangles which are on the same base and in the same parallels equal one another. // || Let //ABC// and //DBC// be triangles on the same base //BC// and in the same parallels //AD// and //BC.// I say that the triangle //ABC// equals the triangle //DBC.// ||  ||  || Produce //AD// in both directions to //E// and //F.// Draw //BE// through //B// parallel to //CA,// and draw //CF// through //C// parallel to //BD.// ||  [|Post.2] [|I.31]  || Then each of the figures //EBCA// and //DBCF// is a parallelogram, and they are equal, for they are on the same base //BC// and in the same parallels //BC// and //EF.// ||  [|I.35]  || Moreover the triangle //ABC// is half of the parallelogram //EBCA,// for the diameter //AB// bisects it. And the triangle //DBC// is half of the parallelogram //DBCF,// for the diameter //DC// bisects it. ||  [|I.34]  || <span style="font-family: "Century Gothic","sans-serif"; color: black">Therefore the triangle //ABC// equals the triangle //DBC.// || <span style="font-size: 10pt; font-family: "Century Gothic","sans-serif"; color: black"> [|C.N] <span style="font-family: "Century Gothic","sans-serif"; color: black"> || <span style="font-family: "Century Gothic","sans-serif"; color: black">Therefore //triangles which are on the same base and in the same parallels equal one another.// || <span style="font-size: 10pt; font-family: "Century Gothic","sans-serif"; color: black">Q.E.D. <span style="font-family: "Century Gothic","sans-serif"; color: black"> ||

<span style="font-size: 8pt; font-family: "Century Gothic","sans-serif"; color: black">In this proposition the triangles have the same base while in the next one the triangles have equal bases. Since the proofs are the same except that this depends on I.35 while the next depends on I.36, and the next is more general, there is no purpose to include this proposition. <span style="font-size: 8pt; font-family: "Century Gothic","sans-serif"; color: black">The justification of the last conclusion is missing. From the statement that the doubles of two magnitudes are equal, we want to conclude that the magnitudes themselves are equal. Although Euclid included no such common notion, others inserted it later. See the commentary on [|Common Notions] for a proof of this halving principle based on other properties of magnitudes.

<span style="font-size: 8pt; font-family: "Century Gothic","sans-serif"; color: black">Use of Proposition 37
<span style="font-size: 8pt; font-family: "Century Gothic","sans-serif"; color: black">This proposition is used in [|I.39], [|I.41] , and [|VI.2]. <span style="font-size: 8pt; font-family: "Century Gothic","sans-serif"; color: black">