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The area (A) of a triangle with base (b) and height (h) is given by $A = \frac{1}{2}bh$.

Given a circle with a chord PQ and a midpoint M, if two other chords AB and CD pass through M and intersect PQ at X and Y respectively, then M is the midpoint of XY.

The orthocenter of a triangle is the point where the three altitudes, or the lines extended from them, intersect.

The circumference (C) of a circle with radius (r) is given by $C = 2\pi r$.

The angle bisector of an angle in a triangle divides the opposite side into two segments that are proportional to the lengths of the other two sides of the triangle.

For any convex polyhedron, the number of vertices (V), faces (F), and edges (E) satisfy the relationship $V - E + F = 2$.

In every non-equilateral triangle, the orthocenter (H), centroid (G), and circumcenter (O) are collinear, and the centroid is between the orthocenter and the circumcenter on this line, known as the Euler line.

Given a triangle ABC and a transversal line intersecting the extended sides AB and AC at points L and M, and intersecting side BC at N, then Menelaus's Theorem states that the points L, M, and N are collinear if and only if $\frac{BL}{LC} \cdot \frac{CM}{MA} \cdot \frac{AN}{NB} = 1$.

In any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle, known as the Morley triangle.

In a circle, the central angle subtended by two points on the circle is twice the inscribed angle subtended by those points.

If equilateral triangles are constructed on the sides of any triangle (either all outwardly or all inwardly), the centers of those equilateral triangles themselves form an equilateral triangle.

For any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side.

In a triangle, three cevians (lines drawn from each vertex to the opposite side) are concurrent (intersect at a single point) if and only if $\frac{AF}{FB} \cdot \frac{BD}{DC} \cdot \frac{CE}{EA} = 1$, where the points D, E, and F lie on the sides of the triangle.

The area (A) of a triangle with sides a, b, and c and semi-perimeter (s) is given by $A = \sqrt{s(s-a)(s-b)(s-c)}$, where $s = \frac{a+b+c}{2}$.

There exists a circle in any non-degenerate triangle that passes through the midpoint of each side, the foot of each altitude, and the midpoint of the line segment from each vertex to the orthocenter.

The area (A) of a circle with radius (r) is given by $A = \pi r^2$.

Given a triangle ABC, and a point P on its circumcircle, the feet of the perpendiculars dropped from P to the sides of ABC (or their extensions) are collinear, and this line is known as the Simson line of point P.

The incenter of a triangle is the point where the three angle bisectors of the triangle intersect, and it is the center of the inscribed circle (incircle).

The circumcenter of a triangle is the point where the perpendicular bisectors of the sides of the triangle intersect, and it is the center of the circumscribed circle (circumcircle).

In a right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). i.e., $a^2 + b^2 = c^2$.

For a triangle with sides a, b, and c and a cevian of length d dividing side a into segments of length m and n, Stewart’s Theorem is given by $b^2m + c^2n = a(d^2 + mn)$.

The centroid of a triangle is the point where the three medians of the triangle intersect, and it serves as the triangle's center of mass.

Given a line and a point not on the line, there is exactly one line parallel to the given line that passes through the point.

For any triangle ABC, with the respective lengths opposite to the angles being a, b, and c, and the length from each vertex to the circumcenter being $O_a$, $O_b$, and $O_c$, Carnot's Theorem is the condition $O_a^2 - a^2 = O_b^2 - b^2 = O_c^2 - c^2$.

For a quadrilateral inscribed in a circle (cyclic quadrilateral), if the lengths of the sides are a, b, c, and d, and the lengths of the diagonals are e and f, then Ptolemy’s Theorem states $ac + bd = ef$.

Two triangles are in perspective axially if and only if they are in perspective centrally; which means that the corresponding sides intersect in collinear points, or their extensions do.

In any triangle with sides of length a, b, and c, and the angle opposite of side c being gamma ($\gamma$), the Law of Cosines states $c^2 = a^2 + b^2 - 2ab\cos(\gamma)$.

For any triangle with sides a, b, c and angles alpha ($\alpha$), beta ($\beta$), and gamma ($\gamma$), the Law of Sines states $\frac{a}{\sin(\alpha)} = \frac{b}{\sin(\beta)} = \frac{c}{\sin(\gamma)}$.

For any triangle that is not equilateral, there is a point such that the total distance from the three vertices of the triangle to this point is minimized. This point is known as Fermat's point or Torricelli point.