As shown in the figure, the triangle ACB and the triangle ECD are isosceles right triangles. The vertex a of the triangle ACB is on the De of the hypotenuse of the triangle ECD. Prove that the square of AE + the square of ad = the square of 2Ac. (hint: connect BD)

As shown in the figure, the triangle ACB and the triangle ECD are isosceles right triangles. The vertex a of the triangle ACB is on the De of the hypotenuse of the triangle ECD. Prove that the square of AE + the square of ad = the square of 2Ac. (hint: connect BD)

Proof: connect to DB
1, in isosceles right angles ⊿ ABC and ⊿ BDC
∵∠ECD=∠ACB=90°
And ∠ e = 45 ° (the acute angle of isosceles right triangle is 45 degrees)
■ ∠ ace = ∠ DCB (equal minus equal difference equals)
∵ EC = DC, AC = BC
≌⊿ AEC ≌⊿ BDC
Ψ AE = BD (the corresponding sides of congruent triangles are equal) one
And ∠ e = ∠ CDB = 45 ° (the corresponding angles of congruent triangles are equal)
2, in ⊿ ADB
∵∠ EDC = 45 ° (the acute angle of isosceles right triangle is 45 degrees)
That is: ⊿ ADB is a right triangle
∴BD²+AD²=AB²…… two
From 1 and 2: AE & # 178; + AD & # 178; = AB & # 178 three
3, in the isosceles right angle ⊿ ABC
∵AB²=AC²+BC²=2AC²…… four
From 3 and 4: BD & # 178; + AD & # 178; = 2Ac & # 178;

A point P jumps from point a one unit away from the origin to the origin. For the first time, it jumps to the midpoint A1 of OA, and for the second time, it jumps from point A1 to the midpoint A2 of OA1
If the point A2 jumps to the midpoint A3 of oa2 for the third time, then the distance from the point 0 to the origin o is ()
Write the reason

Let the nth jump distance be CN,
Then, C1 = 1 / 2, C2 = 1 / 4, C3 = 1 / 8,..., CN = 1 / (2 ^ n),
We find that CN is actually an equal ratio sequence with a common ratio of 1 / 2,
However, C1 + C2 + C3 +... + CN = 1 / 2 [1 - (1 / 2) ^ n] / (1-1 / 2) = 1 - (1 / 2) ^ n,
So the distance from the point 0 to the origin o = (1 / 2) ^ n

Given that point a (a, b), O is the origin, OA = OA1, OA is perpendicular to OA1, then the coordinates of point A1 are

(B, - a) or (- B, a)

A particle P jumps to the origin from a which is one unit of length from the origin
A particle P jumps to the origin direction from point a which is one unit from the origin O1. For the first time, it jumps to the midpoint A1 of OA, for the second time, it jumps from point A1 to the midpoint A2 of OA1, and for the third time, it jumps from point A2 to the midpoint A3 of oa2. If it keeps jumping, the distance from the particle to the origin o after the nth jump is -

When the first jump is finished, the distance from the origin is 1 / 2
After the second jump, the distance from the origin is 1 / 2 square
After the third jump, the distance from the origin is 1 / 2 cubic meters
When the n-th jump is finished, the distance from the origin is 1 / 2 of the n-th power