If circles are drawn taking two sides of a triangle as diameters, prove that the point of intersection of these circles lie on the third side.
Solution:
ABC is a triangle. Taking side AC as diameter, we draw a circle and taking BC as diameter we draw another circle.
We need to prove that point D lies on the third side AB of
.
AC is the diameter which means that
(1)
(Angle in a semi-circle is equal to
)
Similarly, BC is the diameter which means that
(2)
(Angle in a semi-circle is equal to
)
From (1) and (2), we get
which means that
and
form a linear pair.
Therefore, points A, D and B are present on the same straight line.
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