Geometrical: Kaleidoscope by Boris Pritsker

Author:Boris Pritsker
Language: eng
Format: epub
Publisher: Dover Publications, Inc.
Published: 2017-09-12T04:00:00+00:00

Figure 6

Solution. Let us first prove an auxiliary statement:

The areas of the triangles AOB and COD are equal.

Triangles ABD and ACD have the common base AD and equal altitudes dropped to it. Therefore, their areas are equal. Each of the triangles ABD and ACD consists of a common part – triangle AOD and the triangles AOB and COD respectively. Then the area of AOB must equal to the area of COD.

Our goal will be to express the area of AOB or COD in terms of S′ and S″. If we manage to do it, the last step in finding the area of ABCD will be just to add the areas of the triangles AOB, BOC, COD, and DOA.

From the similar triangles BOC and DOA we get , or, expressing OA in terms of CO,

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