A Proof of Ceva’s Theorem
By
Ceva’s
Theorem:
Case 1: T
lies inside
Let D
lie on
Figure 1:
Then, we have
that:
However,
and since
we obtain
Likewise, for
side
but since,
and since,
we have
Furthermore,
for side
but since,
and since,
we have
Therefore, we
have
Conversely,
assume that
For each point,
M on
Further, the
line
Also,
Let T be the
point of intersection of
and we have
so T
lies on
Case 2: Without
loss of generality, let T lie on the opposite side of
Let D
lie outside of
Figure 2:
Then, we have,
for side
However, since
and since
we have
Likewise, for
side
However, since
and since
we have
Also, for side
However, since
and since
we have
Therefore, we
have the equation
Conversely,
assume that
then it is also
true that
For each point,
M on
Further, the
line
Also,
Let T be the
point of intersection of
and we have
so T
lies on
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