Triangle perimeters inequalities

[ins-]

It is given acute angled triangle $ABC$. Let $H_1, H_2, H_3$ are the feets of the altitudes, $T_1, T_2, T_3$ are the tangent points of incircle with the sides, $L_1, L_2, L_3$ are the intersection points of the internal angle bisectors with the sides. Prove that: $P_{H_1H_2H_3} \leq P_{T_1T_2T_3} \leq P_{L_1L_2L_3} \leq \frac{1}{2}P_{ABC}$.

Bonus problem:

Prove that in any triangle $ABC$ is also true: $P_{J_1J_2J_3} \geq 1/2P_{ABC}$. ($J1,J2,J3$) are the tangent points of the excircles with the triangle sides.