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Reference.org
Tangential triangle
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<p>In <a href="/facts/Geometry/wTQf5ffQ">geometry</a>, the tangential triangle of a reference <a href="/facts/Triangle/cRXUwsMn">triangle</a> (other than a <a href="/facts/Right_triangle/dD6M9GAn">right triangle</a>) is the triangle whose sides are on the <a href="/facts/Tangent_line/sF0jH3EI">tangent lines</a> to the reference triangle's <a href="/facts/Circumcircle/RafYT4am">circumcircle</a> at the reference triangle's <a href="/facts/Vertex_(geometry)/ID3cea9z">vertices</a>. Thus the <a href="/facts/Incircle/x9QCn9ML">incircle</a> of the tangential triangle coincides with the circumcircle of the reference triangle. </p><p>The <a href="/facts/Circumcenter/RafYT4am">circumcenter</a> of the tangential triangle is on the reference triangle's <a href="/facts/Euler_line/SKtpSWpb">Euler line</a>,: p. 104, p. 242 as is the <a href="/facts/Center_of_similitude/yyU7NcTl">center of similitude</a> of the tangential triangle and the <a href="/facts/Orthic_triangle/z9H8fFMz">orthic triangle</a> (whose vertices are at the feet of the <a href="/facts/Altitude_(triangle)/z9H8fFMz">altitudes</a> of the reference triangle).: p. 447 : p. 102 </p><p>The tangential triangle is <a href="/facts/Homothetic_transformation/WUmuRuzC">homothetic</a> to the <a href="/facts/Orthic_triangle/z9H8fFMz">orthic triangle</a>.: p. 98 </p><p>A reference triangle and its tangential triangle are in <a href="/facts/Perspective_(geometry)/IKUbVuYw">perspective</a>, and the axis of perspectivity is the <a href="/facts/Lemoine_axis/sMBcSmaz">Lemoine axis</a> of the reference triangle. That is, the lines connecting the vertices of the tangential triangle and the corresponding vertices of the reference triangle are <a href="/facts/Concurrent_lines/zPT6S4w9">concurrent</a>.: p. 165 The center of perspectivity, where these three lines meet, is the <a href="/facts/Symmedian_point/2MWFKJSg">symmedian point</a> of the triangle. </p><p>The tangent lines containing the sides of the tangential triangle are called the <a href="/facts/Exsymmedian/dq1nyC2h">exsymmedians</a> of the reference triangle. Any two of these are concurrent with the third <a href="/facts/Symmedian/CVeo0k45">symmedian</a> of the reference triangle.: p. 214 </p><p>The reference triangle's circumcircle, its <a href="/facts/Nine-point_circle/0UfSxFX9">nine-point circle</a>, its <a href="/facts/Polar_circle_(geometry)/32DR5Cej">polar circle</a>, and the circumcircle of the tangential triangle are <a href="/facts/Coaxal_circles/ovJbT52E">coaxal</a>.: p. 241 </p><p>A right triangle has no tangential triangle, because the tangent lines to its circumcircle at its acute vertices are parallel and thus cannot form the sides of a triangle. </p><p>The reference triangle is the <a href="/facts/Gergonne_triangle/x9QCn9ML">Gergonne triangle</a> of the tangential triangle. </p>