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In geometry, symmedians are three particular lines associated with every triangle. They are constructed by taking a median of the triangle (a line connecting a vertex with the midpoint of the opposite side), and reflecting the line over the corresponding angle bisector (the line through the same vertex that divides the angle there in half). The angle formed by the symmedian and the angle bisector has the same measure as the angle between the median and the angle bisector, but it is on the other side of the angle bisector. The three symmedians meet at a triangle center called the Lemoine point. Ross Honsberger has called its existence "one of the crown jewels of modern geometry". Many times in geometry, if we take three special lines through the vertices of a triangle, or cevians, then their reflections about the corresponding angle bisectors, called isogonal lines, will also have interesting properties. For instance, if three cevians of a triangle intersect at a point P, then their isogonal lines also intersect at a point, called the isogonal conjugate of P. The symmedians illustrate this fact. In the diagram, the medians (in black) intersect at the centroid G. Because the symmedians (in red) are isogonal to the medians, the symmedians also intersect at a single point, L. This point is called the triangle's symmedian point, or alternatively the Lemoine point or Grebe point. The dotted lines are the angle bisectors; the symmedians and medians are symmetric about the angle bisectors (hence the name "symmedian.") Let △ABC be a triangle. Construct a point D by intersecting the tangents from B and C to the circumcircle. Then AD is the symmedian of △ABC. first proof. Let the reflection of AD across the angle bisector of ∠BAC meet BC at M'. Then: second proof. Define D' as the isogonal conjugate of D. It is easy to see that the reflection of CD about the bisector is the line through C parallel to AB. The same is true for BD, and so, ABD'C is a parallelogram.

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Triangle center

In geometry, a triangle center or triangle centre is a point in the triangle's plane that is in some sense in the middle of the triangle. For example, the centroid, circumcenter, incenter and orthocenter were familiar to the ancient Greeks, and can be obtained by simple constructions. Each of these classical centers has the property that it is invariant (more precisely equivariant) under similarity transformations.

Circumcircle

In geometry, the circumscribed circle or circumcircle of a triangle is a circle that passes through all three vertices. The center of this circle is called the circumcenter of the triangle, and its radius is called the circumradius. The circumcenter is the point of intersection between the three perpendicular bisectors of the triangle's sides, and is a triangle center. More generally, an n-sided polygon with all its vertices on the same circle, also called the circumscribed circle, is called a cyclic polygon, or in the special case n = 4, a cyclic quadrilateral.

Symmedian

Explores the mediatrix and bisector properties in right-angled triangles through proofs and constructions.

Covers the geometry of the triangle, including inequalities, heights, and centers.

Covers examples of determining the area of a triangle and finding equations.