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Thus, from this, that the formula is right at n=k, follows that it is faithful and at n=k + This statement is fair at any natural value k. So, the second condition of the principle of mathematical induction too is satisfied. The formula is proved.

Proof: if in a polygon of acute angles more than three, quantity of the obtuse angles adjacent to them (and taken on one at tops will be also more than three. In this case the sum of all adjacent corners taken on one at top for this polygon will be more than 360 °. It is known that at a convex polygon this sum is equal 360 °, therefore this polygon – not convex.

For use in the solution of a formula (*) we will enter an auxiliary piece – height of OD of a triangle of ACD which length we will designate for x. Then length of height of OB of a triangle of ABC will be equal (d2 – x). Let's calculate the area of a quadrangle of ABCD now:

Of course, tasks where obviously it is required to prove falsehood of some statement, but sometimes, for example after promotion of a hypothesis seldom meet, it is easier to try to disprove it through a counterexample, and then, in case of failure, to start proving, than at once to start the proof.

Using equality (and that  1 =  2 (on a condition), we will receive that BCD triangle isosceles, and, therefore, BC = CD. Using the received conclusion and equality (we prove that AB = BC, from where follows the validity of the statement of a task.

Characteristic of a method. By means of some additional construction (extension of a piece, geometrical transformation, etc.) receive a triangle which gives the chance to receive the solution of a task. Usually such triangle possesses two properties, important for the solution of a task:

The analysis – logical reception, the research method consisting that the studied object mentally (or practically) breaks into components (signs, properties, the relations) each of which is investigated separately as part dismembered whole.

Understand the next way of the proof as method of mathematical induction. If it is required to prove the validity of the offer A (n) for all natural n, first, it is necessary to check the validity of the statement And (and, secondly, having assumed the validity of the statement And (k), to try to prove that the statement And (k + is true. If it manages to be proved, and the proof remains fair for each natural value k, according to the principle of mathematical induction the offer A (n) admits true for all values n.

The most natural application of a method of mathematical induction in geometry, close to use of this method in the theory of numbers and in algebra, is an application to the solution of geometrical tasks on calculation. Let's review an example.

If, that, and in the left part of an inequality (we have work of two positive numbers. If, that, and in the left part of an inequality (we have work of two negative numbers. In both cases an inequality (fairly.

Our assumption concerning existence of four (and as shows the analysis of reasonings and bigger quantities of acute angles incorrectly. Therefore, the maximum quantity of acute angles of a convex n-square – three.

Let's notice that this decision was passed for an acute triangle. In case of an obtusangular triangle the result will not change, difference will be only in an initial ratio for the area of S = SABD – SBCD.