Proof
Lay the equiangular triangles side by side so that one pair of equal
angles coincides; the remaining vertices and bases yield a
parallelogram by I.28. Apply VI.2 to the new figure to derive the
proportionality of the sides.
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Depends on (3)
- I.28Proposition I.28If a straight line falling on two straight lines make the exterior angle equal to the interior and opposite angle on…
- I.32Proposition I.32In any triangle, if one of the sides be produced, the exterior angle is equal to the two interior and opposite angles,…
- VI.2Proposition VI.2If a straight line be drawn parallel to one of the sides of a triangle, it will cut the sides of the triangle…
Required by (dependents) (8)
- VI.5Proposition VI.5If two triangles have their sides proportional, the triangles will be equiangular and will have those angles equal…
- VI.6Proposition VI.6If two triangles have one angle equal to one angle and the sides about the equal angles proportional, the triangles…
- VI.7Proposition VI.7If two triangles have one angle equal to one angle, the sides about other angles proportional, and the remaining angles…
- VI.8Proposition VI.8If in a right-angled triangle a perpendicular be drawn from the right angle to the base, the triangles adjoining the…
- VI.18Proposition VI.18On a given straight line to describe a rectilineal figure similar and similarly situated to a given rectilineal figure.
- VI.21Proposition VI.21Figures which are similar to the same rectilineal figure are also similar to one another.
- VI.24Proposition VI.24In any parallelogram the parallelograms about the diameter are similar both to the whole and to one another.
- XIII.8Proposition XIII.8If in an equilateral and equiangular pentagon straight lines subtend two adjacent angles, they cut one another in…
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