One and only one straight line is perpendicular to the known straight line.

Of all the line segments connecting the outer point and the point on the line, the vertical line segment is the shortest.

Parallel axiom

After passing a point outside the straight line, there is one and only one straight line parallel to this straight line.

If both lines are parallel to the third line, then the two lines are also parallel to each other.

The isosceles angles are equal and the two straight lines are parallel.

Internal dislocation angles are equal and two straight lines are parallel.

The internal angles on the same side are complementary and the two straight lines are parallel.

Two straight lines are parallel and have the same angle.

Two straight lines are parallel and have equal internal angles.

These two lines are parallel and complementary.

The line segment between two points is the shortest.

Inference 1

A straight line passing through the midpoint of one waist of the trapezoid and parallel to the bottom will bisect the other waist.

Inference 2

A straight line passing through the midpoint of one side of a triangle and parallel to the other side will be equally divided.

Trilaterality

Triangle midline theorem

The center line of the triangle is parallel to and equal to the third side.

Half of

Trapezoidal midline theorem

The center line of the trapezoid is parallel to the two bottom sides and is equal to the sum of the two bottom sides.

one half

L=(a+b)÷2

S = length × height

Proportional theorem of parallel line segment

Three parallel lines cut two straight lines, corresponding.

The line segments are proportional.

reason

A straight line parallel to one side of a triangle cuts the other two sides (or extension lines on both sides), and the corresponding line segments are proportional.

theorem

If the corresponding line segments obtained by cutting two sides of a triangle (or the extension lines of two sides) are proportional, then this line is parallel to the third side of the triangle.

A straight line parallel to one side of a triangle and intersecting with the other two sides. The three sides of the triangle are proportional to the original three sides.

theorem

A straight line parallel to one side of a triangle intersects with the other two sides (or extension lines of both sides) to form a triangle similar to the original triangle.

The locus of points whose distance is equal to the two ends of a known line segment is perpendicular to the line segment.

bisector

The locus of a point with equal distance to two parallel lines is parallel to these two parallel lines and at a distance.

A straight line of equality

theorem

Three points that are not on the same straight line determine a circle.

① the straight line l intersects with ⊙ o.

dr

Tangent judgment theorem

The straight line passing through the outer end of the radius and perpendicular to the radius is the tangent of the circle.

Property theorem of tangent line

The tangent of a circle is perpendicular to the radius passing through the tangent point.

Tangent secant theorem

The tangent and secant of the circle are drawn from a point outside the circle, and the length of the tangent is the tangent from that point to the tangent.

The proportional average of the lengths of two straight lines at the intersection of a straight line and a circle.

reason

Draw two secants of a circle from a point outside the circle, and the product of the lengths of the two lines from that point to the intersection of each secant and the circle is equal.