Butterfly theorem: P is the midpoint of the chord AB of the circle O. If the two chords CD and EF of the circle O are introduced through the point P, the connecting line DE passes through AB in M and CF passes through AB in N, then MP=NP.
Pappus Theorem: If the vertices of hexagonal ABCDEF are alternately distributed on two straight lines A and B, then the intersections X, Y and Z of the straight lines where its three pairs of opposite sides are located are on the same straight line.
Gauss line theorem: In quadrilateral ABCD, if straight line AB and straight line CD intersect at E, and straight line BC and straight line AD intersect at the midpoint of F, M, N and Q, respectively, AC, BD and EF, then there are M, N and O*** lines.
Mill Theorem: Three corners of a triangle have six bisectors, and the intersection of two adjacent bisectors (not at the same angle) is the vertex of an equilateral triangle.
Napoleon's theorem: If each side of a triangle is an equilateral triangle outward, then their centers will form an equilateral triangle.
Pascal's Theorem: If a hexagon is inscribed on a conic curve, the intersection points of three pairs of opposite sides of the hexagon are on a straight line.
Brian's double theorem: Let a hexagon circumscribe a conic curve, then the connecting line of its three pairs of vertices is a * * * point.
Menelaus Theorem: If a straight line intersects with the sides BC, CA and AB of the triangle ABC at L, M and N respectively, there is: (an/nb) * (bl/LC) * (cm/ma) =1(considering the direction of the line segment, the right side of the equation is-1).
Its inverse theorem: If there are three points L, M and N on the sides BC, CA and AB of triangle ABC or its extension line (at least one point is on the extension line), and (an/nb) * (bl/LC) * (cm/ma) =1,then these three points L, M and N are * * lines.
Seva Theorem: Let O be any point in the triangle ABC, and AO, BO and CO intersect at D, E and F respectively, then (BD/DC) * (CE/EA) * (AF/FB) =1.
Its inverse theorem: Take a point D, E and F on the straight lines BC, CA and AB where the three sides of the triangle ABC are located. If (BD/DC) * (CE/EA) * (AF/FB) =1,then AD, BE and CE are parallel or * * * points.
Stewart Theorem: In triangle ABC, if D is a point on BC, and BD=p, DC=q, AB=c and AC=b, then AD 2 = [(b * b * p+c * c * q)/(p+q)]-PQ.
Taber Theorem: Take the sides of a parallelogram as the sides of a square and make four squares (inside and outside the parallelogram). The quadrangle formed by the center point of a square is a square; Take two adjacent sides of a square as triangle sides and make two equilateral triangles (inside and outside the square). The vertices of these two triangles that are not on the side of the square and the only vertex among the four vertices of the square that is not the vertex of the triangle form an equilateral triangle; Given any point m on an arbitrary triangle ABC, BC, draw two circles, both of which are tangent to AM, BC and the circumscribed circle, and the centers of the two circles are inscribed with a straight line * * *.
Van oberth Theorem: Given a quadrilateral, construct a square outside its edge. Connect the centers of opposite squares to get two line segments. The lengths of the line segments are equal and vertical (Van aubert theorem applies to concave quadrangles).
Sim's Theorem: The necessary and sufficient condition for drawing a perpendicular from a point to three sides of a triangle is that the point falls on the circumscribed circle of the triangle.