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Last edited by hbghlyj at 2025-3-21 00:47:00
mathworld:
A geometric theorem related to the Pentagram and also called the Pratt-Kasapi Theorem.
\[
{|V_1W_1|\over |W_2V_3|}{|V_2W_2|\over |W_3V_4|}{|V_3W_3|\over|W_4V_5|}
{|V_4W_4|\over |W_5V_1|}{|V_5W_5|\over |W_1V_2|}=1\]
\[
{|V_1W_2|\over |W_1V_3|}{|V_2W_3|\over |W_2V_4|}{|V_3W_4|\over|W_3V_5|}
{|V_4W_5|\over |W_4V_1|}{|V_5W_1|\over |W_5V_2|}=1.
\]
In general, it is also true that
\[
{|V_iW_i|\over|W_{i+1}V_{i+2}|}={|V_iV_{i+1}V_{i+4}|\over|V_iV_{i+1}V_{i+2}V_{i+4}|}
{|V_iV_{i+1}V_{i+2}V_{i+3}|\over|V_{i+2}V_{i+3}V_{i+1}|}.
\]
This type of identity was generalized to other figures in the plane and their duals by Pinkernell (1996). |
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