Reuschle's theorem

Reuschle's theorem:
cevians $AP_{a}$, $AP_{b}$ and $AP_{c}$ intersect in $D$
$AP'_{a}$, $AP'_{b}$ and $AP'_{c}$ intersect in $D'$

In elementary geometry, Reuschle's theorem describes a property of the cevians of a triangle intersecting in a common point and is named after the German mathematician Karl Gustav Reuschle (1812–1875).

In a triangle $ABC$ with its three cevians intersecting in a common point other than the vertices $A$, $B$ or $C$ let $P_{a}$, $P_{b}$ and $P_{c}$ denote the intersections of the (extended) triangle sides and the cevians. The circle defined by the three points $P_{a}$, $P_{b}$ and $P_{c}$ intersects the (extended) triangle sides in the (additional) points $P'_{a}$, $P'_{b}$ and $P'_{c}$. Reuschle's theorem now states that the three new cevians $AP'_{a}$, $BP'_{b}$ and $CP'_{c}$ intersect in a comon point as well.

References

• Friedrich Riecke (ed.): Mathematische Unterhaltungen. Volume I, Stuttgart 1867, (reprint Wiesbaden 1973), ISBN 3-500-26010-1, p. 125 (German)