I have recorded and posted to http://www.unizor.com some material about infinity. More precisely, about different infinities. I've placed it into a new topic "Quantity" in the Math Concepts menu item. There are a few mini-theorems and exam problems as well.

Welcome to Unizor, the road to infinity!

## Sunday, November 27, 2011

## Wednesday, November 16, 2011

### Problems in Geometry Series 3

I have uploaded a new lecture to http://www.unizor.com/ with the following geometrical problems.

1. Prove that in an equilateral triangle sum of distances of any internal point to all three sides is constant and is equal to an altitude of a triangle.

2. Prove that a parallelogram with congruent diagonals is a rectangle.

3. Prove that a parallelogram with perpendicular diagonals is a rhombus.

4. Prove that a parallelogram with a diagonal being an angle bisector is a rhombus.

5. Given a rhombus. From a point of intersection of its diagonals we dropped perpendiculars to all four sides. Prove that points of intersection of these perpendiculars with sides form a rectangle.

6. Prove that angle bisectors of a rectangle form a square.

7. Given a square

8. Given a square

9. What condition should a quadrangle satisfy, if a new quadrangle with vertices at midpoints of each its side form

(a) parallelogram,

(b) rhombus,

(c) rectangle,

(d) square.

1. Prove that in an equilateral triangle sum of distances of any internal point to all three sides is constant and is equal to an altitude of a triangle.

2. Prove that a parallelogram with congruent diagonals is a rectangle.

3. Prove that a parallelogram with perpendicular diagonals is a rhombus.

4. Prove that a parallelogram with a diagonal being an angle bisector is a rhombus.

5. Given a rhombus. From a point of intersection of its diagonals we dropped perpendiculars to all four sides. Prove that points of intersection of these perpendiculars with sides form a rectangle.

6. Prove that angle bisectors of a rectangle form a square.

7. Given a square

**. Let points***ABCD***,***A'***,***B'***and***C'***be midpoints of sides***D'***,***CD***,***DA***and***AB***correspondingly. Prove that segments***BC***,***AA'***,***BB'***and***CC'***form a square with each side having a length of 2/5 of the length of each of these segments.***DD'*8. Given a square

**. Let points***ABCD***,***A'***,***B'***and***C'***be positioned on sides***D'***,***AB***,***BC***and***CD***correspondingly, such that segments***DA***,***AA'***,***BB'***and***CC'***are congruent. Prove that***DD'***is a square.***A'B'C'D'*9. What condition should a quadrangle satisfy, if a new quadrangle with vertices at midpoints of each its side form

(a) parallelogram,

(b) rhombus,

(c) rectangle,

(d) square.

### Problems in Geometry Series 2

I have uploaded a new lecture to http://www.unizor.com/ with the following problems.

1. Prove that a median to a hypotenuse in the right triangle is equal to half of that hypotenuse.

Hint: extend the median through hypotenuse by its length.

2. Prove a converse theorem, that if a median in a triangle is equal to half a side it falls on, then an angle, where it starts, is the right angle.

3. Prove that in a right triangle a median and an altitude to its hypotenuse form an angle, equal to difference between this triangle's acute angles.

Hint: use a previous theorem.

4. Given triangle Δ

5. Given an angle ∠

6. Prove that any segment, that connects two bases of a trapezoid, is divided by a median in two congruent parts.

7. Given a triangle Δ

8. Straight lines are drawn through all three vertices of a triangle, forming another (bigger) triangle. Prove that this bigger triangle is divided by sides of a smaller one into 4 triangles, each congruent to a small triangle and each side of a bigger triangle is twice as big as parallel to it side of a smaller triangle.

9. Prove that in an isosceles triangle sum of two distances from any point on a base to two legs is constant and is equal to an altitude from any of two end points of a base to a leg.

10. Change a condition of a previous theorem to use a point on continuation of a base (outside of a triangle). Formulate a theorem in this case and prove it.

1. Prove that a median to a hypotenuse in the right triangle is equal to half of that hypotenuse.

Hint: extend the median through hypotenuse by its length.

2. Prove a converse theorem, that if a median in a triangle is equal to half a side it falls on, then an angle, where it starts, is the right angle.

3. Prove that in a right triangle a median and an altitude to its hypotenuse form an angle, equal to difference between this triangle's acute angles.

Hint: use a previous theorem.

4. Given triangle Δ

**. Segment***ABC***bisects angle ∠***AD***(point***BAC***lies on side***D***). Straight line through point***BC***is parallel to side***D***, intersecting side***AC***at point***AB***. Straight line through point***E***is parallel to side***E***, intersecting side***BC***at point***AC***. Prove that segments***F***and***AE***are congruent.***CF*5. Given an angle ∠

**. Inside it an angle ∠***MXN***is positioned in such a way that***PYQ***is parallel to***MX***,***PY***is parallel to***NX***and distance between***QY***and***MX***is equal to distance between***PY***and***NX***. Prove that a bisector of angle ∠***QY***is a bisector of angle ∠***MXY***.***PYQ*6. Prove that any segment, that connects two bases of a trapezoid, is divided by a median in two congruent parts.

7. Given a triangle Δ

**. Let vertex***ABC***be at the top and side***B***be a base. Let point***AC***be an intersection of two bisectors of angles at the base. Straight line through point***X***parallel to a base intersects its left side***X***and right side***AB***at points***BC***and***M***correspondingly. Prove that segment***N***is equal to a sum of segments***MN***and***AM***.***CN*8. Straight lines are drawn through all three vertices of a triangle, forming another (bigger) triangle. Prove that this bigger triangle is divided by sides of a smaller one into 4 triangles, each congruent to a small triangle and each side of a bigger triangle is twice as big as parallel to it side of a smaller triangle.

9. Prove that in an isosceles triangle sum of two distances from any point on a base to two legs is constant and is equal to an altitude from any of two end points of a base to a leg.

10. Change a condition of a previous theorem to use a point on continuation of a base (outside of a triangle). Formulate a theorem in this case and prove it.

### Problems in Geometry Series 1

I have uploaded a new lecture to http://www.unizor.com/ with the following geometrical problems.

1. Construct a triangle by two sides and a median to the third side.

2. Construct a quadrangle by all four sides and a median segment connecting midpoints of opposite sides.

Hint: use parallel shift.

3. Construct a trapezoid by one of its interior angles, two diagonals and a median (segment between midpoints of two non-parallel sides).

Hint: use parallel shift.

4. Construct a quadrangle by three sides and two interior angles adjacent to one of known sides.

Hint: use parallel shift.

5. Construct a trapezoid by its two bases and two legs.

Hint: use parallel shift.

6. Construct a quadrangle by its four sides, provided one diagonal bisects one angle it is connected to.

Hint: use symmetry relative to an axis.

7. A billiard table has a rectangular shape. There are two balls on it in some position. Determine direction of the movement of one ball in such a way, that, reflecting from all four sides, it will hit the second ball.

Hint: use symmetry relative to table borders on each side.

1. Construct a triangle by two sides and a median to the third side.

2. Construct a quadrangle by all four sides and a median segment connecting midpoints of opposite sides.

Hint: use parallel shift.

3. Construct a trapezoid by one of its interior angles, two diagonals and a median (segment between midpoints of two non-parallel sides).

Hint: use parallel shift.

4. Construct a quadrangle by three sides and two interior angles adjacent to one of known sides.

Hint: use parallel shift.

5. Construct a trapezoid by its two bases and two legs.

Hint: use parallel shift.

6. Construct a quadrangle by its four sides, provided one diagonal bisects one angle it is connected to.

Hint: use symmetry relative to an axis.

7. A billiard table has a rectangular shape. There are two balls on it in some position. Determine direction of the movement of one ball in such a way, that, reflecting from all four sides, it will hit the second ball.

Hint: use symmetry relative to table borders on each side.

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