...Caesar lived, or that Jesus Christ died and rose again, or those by which we come to be assured that the square of the hypotenuse of a right-angled triangle is equal to the square of the other two sides. But in all such regressions we must at last come back to something...

...(AD = CD) = diff. of sqrs. on AD and D C. 112. Three times the sum of the squares on the .sides of a triangle is equal to four times the sum -of the squares of the lines joining the middle point of .each side with the opposite angle, (See fig. of Example 14.) The...

...angle of a triangle is greater than the squares on the other two sides. 15. Five times the square on the hypotenuse of a right.angled triangle is equal to four times the sum of the squares on the medians drawn to the other two sides. 16. Three times the square on a side of an equilateral...

...4AE2 + 4EF2 +4FB2 = AC2 +BD2 + 4EF2. Prop. 5. — Three times the sum of the squares of the fides of a triangle is equal to four times the sum of the squares of the lines bisecting the sides of the triangle. Dem. — Let D, E, F be the middle points of the sides....

...Pythagoras to this science was not fruitless. To him we owe the discovery of the geometrical truth that the square of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the other two sides. Religion formed the basis of moral action. Pythagoras,...

...4AE2 + 4EP +4FB2 = AC2 +BD2 + 4EE2. Prop. 5. — Three times the sum of the squares of the sides of a triangle is equal to four times the sum of the squares of the lines bisecting the sides of the triangle. Dem. — Let D, E, F be the middle points of the sides....

...may be solved by the use of the two following principles, which are demonstrated in geometry. 1st. The square of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the other two sides. 2d. The areas of two circles are to each other as the...

...given we can define all the angles. Now the relations of trigonometrical ratios to one another (since the square of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the two sides) are as follows : — Since a1 + b' — c\ .. .... . , a2 62...

...nor thirty centuries, affect the clearness, or the charm, of Geometrical truths. Such a theorem as ' the square of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the sides ' is as dazzlingly beautiful now as it was in the day when Pythagoras...

...the diagonals of a quadrilateral is bisected by the lines joining the diameters of the quadrilateral. 4. Prove that five times the square of the hypotenuse...of the squares of the medians from its extremities. PROPOSITION XXIII. 416. Theorem : Of three similar figures constructed on the hypotenuse and the sides...