A math isosceles triangle problem, quick!!

1.The intersection point of the perpendicular bisector and the straight line of ab is point c, then ac=.

2.Draw a circle with ab as the radius and point a as the center of the circle, and there are two intersection points with the straight line, which are c1 and the radius of the circle. 2 pcs.

3.Draw a circle with ab as the radius and point b as the center of the circle, and the two points that intersect with the straight line are c3 and the radius of the circle. 2 pcs.

It has nothing to do with whether ab is parallel to a straight line.

Isosceles triangle ABC

1. An isosceles triangle with AB as the base. The intersection point of the perpendicular bisector of AB and the known straight line is point C.

2. Take ab as an isosceles triangle with a waist.

1) Let ab=ac draw a circle with point A as the center of the circle and the length of ab as the radius, and the circle has an intersection point with a known straight line, and the intersection point is c (if the distance of ab is equal to the distance from a to the straight line, then there is an intersection point; If the ab distance is greater than the distance from a to the straight line, there are two intersections).

2) Order. ab=bc draws a circle with point b as the center and ab length as the radius, the circle has an intersection point with a known straight line, and the intersection point is c (if the distance of ab is equal to the distance from a to the straight line, there is an intersection point; If the ab distance is greater than the distance from b to the straight line, there are two intersections).

Firstly, the classification is discussed: (1) the triangle made with ab as the base of the isosceles triangle and (2) the triangle made by the waist with ab as the isosceles triangle; The case of (2) can also be classified and discussed: point A is the vertex of an isosceles triangle and point B is the vertex of an isosceles triangle.

For more specific, such as drawing a circle, you should understand it if you try to draw it yourself, which is what you need to do in the case of (2) above.

1. Equilateral triangles.

Properties: 1. All three sides are equal;

2. All three corners.

equal, and each angle is equal to 60°;

Judgment: 1. A triangle with equal three sides is an equilateral triangle;

2. There is an isosceles triangle with an angle of 60°.

is an equilateral triangle.

2. Isosceles triangle.

Properties: 1. The two base angles of an isosceles triangle are equal;

2. The height on the bottom edge of the isosceles triangle, the middle line on the bottom edge, and the bisector line at the top angle coincide with each other.

Judgment: 1. A triangle with equal sides is an isosceles triangle;

2. Two triangles with equal angles are isosceles triangles;

3. Right triangles.

Properties: 1. The sum of squares of two right-angled sides is equal to the square of hypotenuse;

2. Two acute angles are redundant;

3. The middle line on the hypotenuse is equal to half of the hypotenuse;

The right-angled edge of the acute angle is equal to half of the hypotenuse.

Judgment: 1. A triangle with an angle of 90° is a right triangle;

2. If a triangle has the sum of the squares of the two sides equal to the square of the third side, then the triangle is a right-angled triangle;

3. If the middle line on one side of a triangle is equal to half of this side, then the triangle is a right triangle.

Proof: ab1 bc

ab1c1 is obtained by abc rotating around point a, and ab=bc ab=bc=ab1=b1c1

bac= c= b1ac1= ac1b1 and b1c1b+ ac1b1+ ac1c=180°, i.e. b1c1b+2 c=180°

and abc+ bac+ c=180°, i.e. abc+2 c=180° abc= b1c1b, abc= ab1c1b, abc1c1= b1c1b

ab1‖bc

Answer: (1) The above two students are not comprehensive, it should be: the size of the other two corners is 75° and 75° or 30° and 120°

Here's why: when a is the top angle, let the bottom angle be

The remaining two corners are 75° and 75°

When a is the bottom angle, let the top angle be , 30°+30°+180°,=120°, and the other two angles are 30° and 120° respectively

2) The feeling is: when solving the problem, the thinking of the problem should be comprehensive, and some topics should be classified and discussed, and the classification should be done without duplication or omission

1) Both cases are true, because it depends on whether the angle of 30° is the bottom angle or the top angle.

2) Look at math problems rigorously.

Because ac=ad

So angular ACD = angular ADC

So the angle acb = the angle ade

And because Sun Zhi is ab=ae, ac=ad

So the triangle ABC triangle is slow to aed

So scrambling bc=de

Not equal. The area of the three small triangles is the same, according to the formula for the area of the triangle: s=(1 2)absinc is obviously unequal.

Certainly not equal, because it has been mathematically proven that it is impossible to divide any angle into threes, and if you do that the three angles are equal means that it is possible to divide any angle into three equal parts, which is obviously impossible.

Unequal middle is larger. Same on both sides.

Definitely not equal It is very difficult to draw thirds with a ruler gauge, and many graduate students will not, do you think it will be so easy to make it???

Equal. Because in an isosceles triangle, the bottom edge is divided into three equal parts, then the angles of the vertices are divided into three equal parts. Remember to adopt it.

Knowledge point 1: The nature of isosceles triangles.

1) Symmetry: The isosceles triangle is axisymmetric figure, the straight line where the middle line on the bottom edge of the isosceles triangle is its axis of symmetry, or the straight line where the height on the bottom edge is its axis of symmetry, or the straight line where the bisector of the apex angle is its axis of symmetry.

2) Three-in-one: The bisector of the top angle of the isosceles triangle, the middle line on the bottom edge, and the height on the bottom edge coincide with each other.

3) Equilateral Equilateral Equal Angles: The two base angles of an isosceles triangle are equal.

Tips: "Three lines in one" means that the corresponding angular bisector, middle line, and high line are actually just one line segment when drawing, that is, a line segment is not only the bisector of the top angle, but also the middle line on the bottom edge, or the height on the bottom edge, which cannot be confused.

Knowledge point 2: The determination theorem of isosceles triangle.

Theorem: If the two angles of a triangle are equal, then the sides opposite the two angles are also equal (abbreviation: equigonal to equiside).

Tips: (1) The two angles in the theorem must be two inner angles in the same triangle, and cannot appear in two triangles; (2) The two sides in the conclusion should be the "opposite edges" of the two inner angles, and this correspondence should not be confused; (3) The function of this theorem is to prove that a triangle is an isosceles triangle.

Knowledge point 3: The nature and determination of equilateral triangles.

1.The three angles of an equilateral triangle are all equal, and each angle is equal to 60°

2.An equilateral triangle has all the properties of an isosceles triangle and has a "three-in-one" on each sideThus an equilateral triangle is an axisymmetric figure that has three axes of symmetry, whereas an isosceles triangle (non-equilateral triangle) has only one axis of symmetry.

3.There is an isosceles triangle with an angle of 60° which is an equilateral triangle.

Extension: An equilateral triangle is a special type of triangle that makes it easy to know that the three heights (or three midlines and three bisectors of the corner) of an equilateral triangle are all equal.

Knowledge point 4: Application of isosceles triangle properties.

In addition to the "three lines in one", there are also special properties between the main line segments in the triangle, such as:

1) The bisector of the two base angles of an isosceles triangle is equal; (2) the midline on both sides of the isosceles triangle is equal;

3) The height on both sides of the isosceles triangle is equal; (4) The distance from the midpoint on the base edge of the isosceles triangle to the two waists is equal.