An elementary treatise on geometry : simplified for beginners not versed in algebra . aDT; because these two triangles are upon thesame base, aD, and between the same parallels, aD, TC ; therefore(by adding to each of them the triangle cAD) the two trianglesADT and aAC are also equal; that is, ADT is also ^ of the trian-gle ABC. In the same manner (by drawing the line 6C) it maybe proved that the triangle BKT is also I of the triangle , the triangles ATD, DTE, ETF, FTG, are, by construc-tion, all equal to one another, having equal bases and heights (seethe demonstration to the last
An elementary treatise on geometry : simplified for beginners not versed in algebra . aDT; because these two triangles are upon thesame base, aD, and between the same parallels, aD, TC ; therefore(by adding to each of them the triangle cAD) the two trianglesADT and aAC are also equal; that is, ADT is also ^ of the trian-gle ABC. In the same manner (by drawing the line 6C) it maybe proved that the triangle BKT is also I of the triangle , the triangles ATD, DTE, ETF, FTG, are, by construc-tion, all equal to one another, having equal bases and heights (seethe demonstration to the last problem); and for the same reason arethe triangles BTK, KTI, ITH equal to one another; therefore eachof the seven triangles ATD, DTE, ETF, FTG, BTK, KTI, ITH,isJ of the triangle ABC ; consequently the quadrilateral GTHCmust be the remaining one eighth of the triangle ABC; and thearea of the triangle ABC is divided into eight equal parts. Problem XXXVL To divide a triangle, from agiven point icithin it, into a given number of equal parts.[This problem is intended for elder pupils.]. Solution. Let ABC be the given triangle, which isto be divided, say, into five equal parts ; T the pointfrom which the lines of division are to be drawn. 1. Through the point T and the vertex A of the trian-gle, draw the line * 174 GEOMETRY. 2. Take any side of the triangle, say BC, and make,when, as here, the triangle is to be divided into five equalparts, BE and CF equal to -i of BC, and draw the linesEe, Fy, parallel to the sides AB, AC ; these lines willmeet the line AT in the points e and/. 3. From T draw the lines TB, TC, to the vertices Band C of the triangle ABC, and from e and/, the lineselJG parallel to TB, TC. 4. Join TI, TG ; then each of the triangles ATI,ATG, is -i of the given triangle ABC. 5. In order to determine the other points of division,it is only necessary to cut off from the sides AB, AC, asmany distances, equal to AI, AG, respectively, as is possible(see the solution of
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Keywords: ., bookauthorgrundfrancisjfrancisjoseph18051863, bookcentury1800
