The big idea of ​​Geometry is to give each point on the plane a pair of numbers. We choose two lines perpendiculars (the axes) and a point is located by the distances $x$ and $y$ to them. Today We call this Cartesian plane, in homage to him (Cartesius was Descartes’ name in Latin).

With coordinates, a figure becomes an equation. See three examples.

A page from *The Geometry* (1637). The letters a, b appear for known values ​​and the powers written at the top.
A page from The Geometry (1637). The letters a, b appear for known values ​​and the powers written at the top.René Descartes · public domain

The distance between two points. The points $P = (x_1, y_1)$ and $Q = (x_2, y_2)$ form, with the point $(x_2, y_1)$, a right triangle of legs $x_2 - x_1$ and $y_2 - y_1$. By the theorem of Pythagoras:

$$d(P, Q) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}.$$
Drag P and Q: the sides of the triangle are Δx and Δy, and the distance comes from the Pythagorean theorem.

The circle. A circle with center $(a, b)$ and radius $r$ is the set of points that are at distance $r$ from the center. Using the formula above and squaring:

$$(x - a)^2 + (y - b)^2 = r^2.$$

Where a straight line cuts a circle. In Greek geometry, this would require a construction with ruler and compass. With Descartes’ idea, it is enough to solve a system. For the circumference $x^2 + y^2 = 25$ and the straight line $y = x + 1$:

$$\begin{aligned} x^2 + (x + 1)^2 &= 25 \\ 2x^2 + 2x - 24 &= 0 \\ x^2 + x - 12 &= 0 \\ (x + 4)(x - 3) &= 0. \end{aligned}$$

The meeting points are $(3, 4)$ and $(-4, -3)$. Check: $3^2 + 4^2 = 25$ and $(-4)^2 + (-3)^2 = 25$.

Change the line and see algebra follow geometry. When $\Delta$ of the 2nd degree equation becomes negative, the straight line passes outside the circle:

The line y = x + b and the circle x² + y² = 25. With b = 1 the points are (3; 4) and (−4; −3).

A proof in three lines. The diagonals of a parallelogram bisect each other. Place a vertex at origin: the vertices are $O = (0, 0)$, $A = (a, 0)$, $B = (b, c)$ and $C = (a + b, c)$. O midpoint of the diagonal $OC$ is $\left(\frac{a+b}{2}, \frac{c}{2}\right)$. The midpoint of the diagonal $AB$ is $\left(\frac{a+b}{2}, \frac{c}{2}\right)$. It’s the same point, so the diagonals cut each other middle. No fancy drawings were necessary: ​​just beads.

Drag A and B: the golden point is the middle of the two diagonals at the same time, whatever the parallelogram.

Pierre de Fermat arrived at the same idea at the same time, independently, but did not publish it. The two They became rivals and even exchanged challenges by letter.

All topics by René Descartes · ← The life of René Descartes · Square root with ruler and compass →

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