Points on a plane: \(A,\) \(B,\) \(C,\) \(D\)

Point coordinates: \(\left( {{x_0},{y_0}} \right),\) \(\left( {{x_1},{y_1}} \right),\) \(\left( {{x_2},{y_2}} \right),\) \(\left( {{x_3},{y_3}} \right)\)

Centroid: \(M\left( {{x_0},{y_0}} \right)\)

Incenter: \(I\left( {{x_0},{y_0}} \right)\)

Circumcenter: \(O\left( {{x_0},{y_0}} \right)\)

Orthocenter: \(H\left( {{x_0},{y_0}} \right)\)

Point coordinates: \(\left( {{x_0},{y_0}} \right),\) \(\left( {{x_1},{y_1}} \right),\) \(\left( {{x_2},{y_2}} \right),\) \(\left( {{x_3},{y_3}} \right)\)

Centroid: \(M\left( {{x_0},{y_0}} \right)\)

Incenter: \(I\left( {{x_0},{y_0}} \right)\)

Circumcenter: \(O\left( {{x_0},{y_0}} \right)\)

Orthocenter: \(H\left( {{x_0},{y_0}} \right)\)

Distance between two points: \(d\)

Real number: \(\lambda\)

Polar angles: \(\varphi,\) \({\varphi_1},\) \({\varphi_2}\)

Polar radii: \(r,\) \({r_1},\) \({r_2}\)

Area: \(S\)

Real number: \(\lambda\)

Polar angles: \(\varphi,\) \({\varphi_1},\) \({\varphi_2}\)

Polar radii: \(r,\) \({r_1},\) \({r_2}\)

Area: \(S\)

- A two-dimensional Cartesian coordinate system is formed by two mutually perpendicular axes. The axes intersect at the point \(O,\) which is called the origin. In the right-handed system, one of the axes (\(x\)-axis) is directed to the right, the other \(y\)-axis is directed vertically upwards. The coordinates of any point on the \(xy\)-plane are determined by two real numbers \(x\) and \(y\), which are orthogonal projections of the points on the respective axes. The \(x\)-coordinate of the point is called the abscissa of the point, and the \(y\)-coordinate is called its ordinate.
- The distance between two points \(A\left( {{x_1},{y_1}} \right)\) and \(B\left( {{x_2},{y_2}} \right)\) on a plane is determined by the expression

\(d = \left| {AB} \right| =\) \( \sqrt {{{\left( {{x_2} – {x_1}} \right)}^2} + {{\left( {{y_2} – {y_1}} \right)}^2}} \) - Dividing a line segment in the ratio \(\lambda\)

Let the point \(C\left( {{x_0},{y_0}} \right)\) divide the segment \(AB\) in the ratio \(\lambda\). The coordinates of the point \(C\) are given by the formulas

\({x_0} = {\large\frac{{{x_1} + \lambda {x_2}}}{{1 + \lambda }}\normalsize},\;\) \({y_0} = {\large\frac{{{y_1} + \lambda {y_2}}}{{1 + \lambda }}\normalsize},\;\) \(\lambda = {\large\frac{{AC}}{{CB}}\normalsize},\;\) \(\lambda \ne – 1, \)

where \({x_1}\), \({y_1}\) is the coordinate of the point \(A\), and \({x_2}\), \({y_2}\) is the coordinate of the point \(B\). - The coordinates of the midpoint of the segment are obtained from the previous formula at \(\lambda = 1\):

\({x_0} = {\large\frac{{{x_1} + {x_2}}}{2}\normalsize},\;\) \({y_0} = {\large\frac{{{y_1} + {y_2}}}{2}\normalsize},\;\) \(\lambda = {\large\frac{{AC}}{{CB}}\normalsize} = 1.\) - The point of intersection of the medians in a triangle has the following coordinates:

\({x_0} = {\large\frac{{{x_1} + {x_2} + {x_3}}}{3}\normalsize},\;\) \({y_0} = {\large\frac{{{y_1} + {y_2} + {y_3}}}{3}\normalsize},\)

where \(A\left( {{x_1},{y_1}} \right)\), \(B\left( {{x_2},{y_2}} \right)\) and \(C\left( {{x_3},{y_3}} \right)\) are the vertices of the triangle \(ABC\). In a triangle with uniform density, the point of intersection of the medians \(M\left( {{x_0},{y_0}} \right)\) is also the center of gravity or centroid. - The coordinates of the point of intersection of the angle bisectors (incenter) of a triangle are given by the relationships:

\({x_0} = {\large\frac{{a{x_1} + b{x_2} + c{x_3}}}{{a + b + c}}\normalsize},\;\)\({y_0} = {\large\frac{{a{y_1} + b{y_2} + c{y_3}}}{{a + b + c}}\normalsize},\)

where \(a = BC\), \(b = AC\), \(c= AB\). - The point of intersection of the perpendicular bisectors of a triangle is the centre of the circumscribed circle (circumcenter) and has the coordinates

\({x_0} = {\large\frac{{\left| {\begin{array}{*{20}{c}} {x_1^2 + y_1^2} & {{y_1}} & 1\\ {x_2^2 + y_2^2} & {{y_2}} & 1\\ {x_3^2 + y_3^2} & {{y_3}} & 1 \end{array}} \right|}}{{2\left| {\begin{array}{*{20}{c}} {{x_1}} & {{y_1}} & 1\\ {{x_2}} & {{y_2}} & 1\\ {{x_3}} & {{y_3}} & 1 \end{array}} \right|}}\normalsize},\;\) \({y_0} = {\large\frac{{\left| {\begin{array}{*{20}{c}} {{x_1}} & {x_1^2 + y_1^2} & 1\\ {{x_2}} & {x_2^2 + y_2^2} & 1\\ {{x_3}} & {x_3^2 + y_3^2} & 1 \end{array}} \right|}}{{2\left| {\begin{array}{*{20}{c}} {{x_1}} & {{y_1}} & 1\\ {{x_2}} & {{y_2}} & 1\\ {{x_3}} & {{y_3}} & 1 \end{array}} \right|}}\normalsize}\) - The point of intersection of the altitudes (orthocenter) of a triangle

\({x_0} = {\large\frac{{\left| {\begin{array}{*{20}{c}} {{y_1}} & {{x_2}{x_3} + y_1^2} & 1\\ {{y_2}} & {{x_3}{x_1} + y_2^2} & 1\\ {{y_3}} & {{x_1}{x_2} + y_3^2} & 1 \end{array}} \right|}}{{\left| {\begin{array}{*{20}{c}} {{x_1}} & {{y_1}} & 1\\ {{x_2}} & {{y_2}} & 1\\ {{x_3}} & {{y_3}} & 1 \end{array}} \right|}}\normalsize},\;\) \({y_0} = {\large\frac{{\left| {\begin{array}{*{20}{c}} {x_1^2 + {y_2}{y_3}} & {{x_1}} & 1\\ {x_2^2 + {y_3}{y_1}} & {{x_2}} & 1\\ {x_3^2 + {y_1}{y_2}} & {{x_3}} & 1 \end{array}} \right|}}{{\left| {\begin{array}{*{20}{c}} {{x_1}} & {{y_1}} & 1\\ {{x_2}} & {{y_2}} & 1\\ {{x_3}} & {{y_3}} & 1 \end{array}} \right|}}\normalsize}\) - Area of a triangle

\(S =\) \(\left( \pm \right){\large\frac{1}{2}\normalsize}\left| {\begin{array}{*{20}{c}} {{x_1}} & {{y_1}} & 1\\ {{x_2}} & {{y_2}} & 1\\ {{x_3}} & {{y_3}} & 1 \end{array}} \right| =\) \( \left( \pm \right){\large\frac{1}{2}\normalsize}\left| {\begin{array}{*{20}{c}} {{x_2} – {x_1}} & {{y_2} – {y_1}}\\ {{x_3} – {x_1}} & {{y_3} – {y_1}} \end{array}} \right|,\)

where \(A\left( {{x_1},{y_1}} \right)\), \(B\left( {{x_2},{y_2}} \right)\) and \(C\left( {{x_3},{y_3}} \right)\) are the vertices of the triangle \(ABC\), and the sign in the right side is chosen so that the area of the triangle is nonnegative. - Area of a quadrilateral

\(S =\) \(\left( \pm \right){\large\frac{1}{2}\normalsize}\big[ {\left( {{x_1} – {x_2}} \right)\left( {{y_1} + {y_2}} \right) }\) \(+\;{ \left( {{x_2} – {x_3}} \right)\left( {{y_2} + {y_3}} \right) }\) \(+\;{ \left( {{x_3} – {x_0}} \right)\left( {{y_3} + {y_0}} \right) }\) \(+\;{ \left( {{x_0} – {x_1}} \right)\left( {{y_0} + {y_1}} \right)} \big], \)

where \(A\left( {{x_1},{y_1}} \right)\), \(B\left( {{x_2},{y_2}} \right)\), \(C\left( {{x_3},{y_3}} \right)\), \(D\left( {{x_0},{y_0}} \right)\) are the vertices of the quadrilateral \(ABCD\). The sign in the right side is chosen so that the area of the quadrilateral is nonnegative. - Distance between two points in polar coordinates

\(d = \left| {AB} \right| =\) \( \big[ {r_1^2 + r_2^2 }\) \(-\;{ 2{r_1}{r_2}\cos \left( {{\varphi_2} – {\varphi_1}} \right)}\big]^{\large{\frac{1}{2}}\normalsize} \) - Converting from rectangular to polar coordinates

\(x = r\cos \varphi ,\;\) \(y = r\sin \varphi \) - Converting from polar to rectangular coordinates

\(r = \sqrt {{x^2} + {y^2}} ,\;\) \(\tan \varphi = {\large\frac{y}{x}\normalsize}.\)